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QUASI-LOCALITY BOUNDS FOR QUANTUM LATTICE SYSTEMS. PART I.

LIEB-ROBINSON BOUNDS, QUASI-LOCAL MAPS, AND SPECTRAL FLOW

AUTOMORPHISMS

BRUNO NACHTERGAELE, ROBERT SIMS, AND AMANDA YOUNG

Abstract. Lieb-Robinson bounds show that the speed of propagation of information under theHeisenberg dynamics in a wide class of non-relativistic quantum lattice systems is essentiallybounded. We review work of the past dozen years that has turned this fundamental result intoa powerful tool for analyzing quantum lattice systems. We introduce a unified framework for awide range of applications by studying quasi-locality properties of general classes of maps definedon the algebra of local observables of quantum lattice systems. We also consider a number ofgeneralizations that include systems with an infinite-dimensional Hilbert space at each lattice siteand Hamiltonians that may involve unbounded on-site contributions. These generalizations requirereplacing the operator norm topology with the strong operator topology in a number of basic resultsfor the dynamics of quantum lattice systems. The main results in this paper form the basis for adetailed proof of the stability of gapped ground state phases of frustration-free models satisfying aLocal Topological Quantum Order condition, which we present in a sequel to this paper.

Contents

1. Introduction 22. Some basic properties of quantum dynamics 42.1. Properties of continuity, measurability, and integration in B(H) 52.2. Dynamical equations and the Dyson series 72.3. The dynamics for a class of unbounded Hamiltonians 113. Lieb-Robinson bounds and infinite volume dynamics of lattice systems 163.1. Lieb-Robinson estimates for bounded time-dependent interactions 173.2. A class of unbounded Hamiltonians 243.3. The infinite-volume dynamics 264. Local approximations 334.1. Local approximations of observables 334.2. Application to quantum lattice models 355. Quasi-local maps 435.1. General quasi-local maps 435.2. Examples of quasi-local maps 455.3. Compositions of quasi-local maps 495.4. Quasi-local transformations of interactions 535.5. Quasi-locality for the difference of two dynamics 626. The spectral flow 646.1. Set up and main results 656.2. An explicit weight 676.3. On weighted integrals of dynamics 706.4. The proof of Theorem 6.3 736.5. Quasi-locality of the spectral flow 75

Date: March 4, 2019.Based upon work supported by the National Science Foundation under Grant DMS-1515850 and DMS-1813149.

1

http://arxiv.org/abs/1810.02428v2

2 B. NACHTERGAELE, R. SIMS, AND A. YOUNG

7. Automorphic equivalence of gapped ground state phases 837.1. Uniformly gapped curves and automorphic equivalence 837.2. Automorphic equivalence with symmetry 877.3. Examples of uniformly gapped curves 908. Appendix 918.1. On F -functions 918.2. On weighted F -functions 938.3. Simple transformations of F -functions 958.4. Basic interaction bounds 96Acknowledgements 101References 102

1. Introduction

Quantum many-body theory comes in two flavors. The first is the relativistic version genericallyreferred to as Quantum Field Theory (QFT), used for particle physics, and the second is non-relativistic many-body theory, which serves as the basic framework for most of condensed matterphysics. The close physical and mathematical similarities between the two have long been recog-nized and exploited with great success. Bogoliubov’s theory of superfluidity and the BCS theory ofsuperconductivity serve as definitive proof that quantum fields are a useful and even fundamentalconcept for understanding non-relativistic many-body systems. Condensed matter theorists havedeveloped field theory techniques that are now omnipresent in the subject [1, 3, 42, 62, 79, 86, 125].See [51,117] for reviews on recent progress in the mathematics of Bogoliubov’s theory and the BSCtheory of superfluidity.

The absence of Lorentz-invariance (and the associated constant speed of light c leading to theall-important property of locality in the sense of Haag [48]) in non-relativistic many-body theoriesis the most obvious difference between the two perspectives. Given the importance of this invari-ance in QFT, which plays an essential role in deriving many of the fundamental properties, andthe strong constraints it imposes on its mathematical structure, one would expect that its absencewould prevent any close analogy between the relativistic and the non-relativistic setting to holdtrue. Contrary to this expectation, successful applications of QFT to problems of condensed matterphysics have been numerous. Quantum Field Theories have provided accurate descriptions as effec-tive theories describing important aspects such as excitation spectra and derived quantities. Thistypically involves a scaling limit of some type. Conformal Field Theories have been spectacularlyeffective in describing and classifying second order phase transitions. Also here a scaling limit isoften implied.

The quasi-locality properties that are the subject of study in this paper partly explain the closer-than-expected similarities between QFT and the non-relativistic many-body theory of condensedmatter systems. More importantly, they make it possible to prove that much of the mathematicalstructure of QFT can be found in non-relativistic many-body systems in an approximate sense.Instead of asymptotic statements and qualitative comparisons, we can prove quantitative estimates:the quasi-locality of the dynamics is characterized by an approximate light-cone with errors thatcan be bounded explicitly. These are the Lieb-Robinson bounds, which have been an essentialingredient in a large number of breakthrough results in the past dozen years.

Although the result of Lieb and Robinson dates back to the early 70’s [75, 112], the impetusfor the recent flurry of activity and major applications came from the work of Hastings on theLieb-Schultz-Mattis Theorem in arbitrary dimension [54]. The possibility of adapting some ofthe major results of QFT to (non-relativistic) quantum lattice systems was anticipated by others.For example, Wreszinski studied the connection between the Goldstone Theorem [73], charges, and

QUASI-LOCALITY BOUNDS I 3

spontaneous symmetry breaking [127]. A rigorous proof of a non-relativistic Exponential ClusteringTheorem, long known in QFT [43,114], did not appear until the works [57,92]. The time-evolutionof quantum spin systems turns local observables into quasi-local ones. Lieb-Robinson bounds wereapplied to approximate such quasi-local observables by strictly local ones with an error boundin [23,88,91]. These (sequences of) strictly local approximations are what is used in many concreteapplications and also have conceptual appeal. Further extensions of Lieb-Robinson bounds and asampling of interesting applications are discussed in Section 3.

Apart from offering a review of the state of the art of quasi-locality estimates, in this paper we alsoextend existing results in the literature in several directions. First, for most of the results we allowthe quantum system at each lattice site to be described by an arbitrary infinite-dimensional Hilbertspace. For many results, the single-site Hamiltonians may be arbitrary densely defined self-adjointoperators. Another generalization in comparison to the existing literature, made necessary by theconsideration of unbounded Hamiltonians, is that time-dependent perturbations are assumed to becontinuous with respect to the strong operator topology instead of the operator norm topology.In order to handle this more general situation, a number of technical issues need to be addressedrelated to the continuity properties of operator-valued functions and of the dynamics generated bystrongly continuous time-dependent interactions. These technical issues cascade through the betterpart of the paper. We will understand if the reader is surprised by the length of the paper, since wewere taken aback ourselves as we were completing the manuscript. Many proofs can be shortenedif one is only interested in particular cases. Indeed, in many cases results for more restricted casesexist in the literature. There are also places, however, where the published results in the literatureprovide only weaker estimates or have incomplete proofs.

This paper has seven sections in addition to this introduction. In Section 2 we review theconstruction and basic properties of quantum dynamics in the context of this work. This includes acareful presentation of analysis with operator-valued functions using the strong operator topology.Section 3 is devoted to Lieb-Robinson bounds and their application to proving the existence ofthe thermodynamic limit of the dynamics. We also derive an estimate on the dependence of thedynamics on the interactions and introduce a notion of convergence of interactions that implies theconvergence of the infinite-volume dynamics. Section 4 is devoted to the approximation of quasi-local observables by strictly local ones by means of suitable maps called conditional expectations.Because they are needed for our applications, the continuity properties of a class of such maps arestudied in detail. A general notion of quasi-local maps is introduced in Section 5, and we study theproperties of several operations involving such maps that are used extensively in applications. InSection 6 we construct an auxiliary dynamics called the spectral flow (also called the quasi-adiabaticevolution), which is the main tool in recent proofs of the stability of spectral gaps and gapped groundstate phases. A first application of the spectral flow is the notion of automorphic equivalence,discussed in Section 7, which allows us to give a precise definition of a gapped ground state phaseas an equivalence class for a certain equivalence relation on families of quantum lattice models.Section 8 is an appendix in which we collect a number of arguments that are used throughout thepaper.

Our original motivation for this work was to supply all the tools needed for the results of [96].However, this work can now be read as a stand-alone review article about quasi-locality estimates forquantum lattice systems. Since the sequel of this paper, [96], will be devoted to applying the quasi-locality bounds and the spectral flow results from this work to prove stability of gapped groundstate phases, the examples and applications here will be chosen in support of the presentation ofthe general results.

Throughout this paper we focus on so-called bosonic lattice systems, for which observables withdisjoint support commute. Virtually all results carry over to lattice fermion systems with only minorchanges. This is discussed in some detail in [97]. Another extension of quasi-locality techniquesnot covered in this paper is the case of so-called extended operators. An important example are


the half-infinite string operators that create the elementary excitations in models with topologicalorder such as the Toric Code model [71]. Lieb-Robinson bounds for such non-quasi-local operatorsare used in [29].

2. Some basic properties of quantum dynamics

In this paper, the primary object of study is the Heisenberg dynamics acting on a suitable algebraof observables for a finite or infinite lattice system. For finite systems, this dynamics is expressedwith a unitary propagator U(t, s), s ≤ t ∈ R, on a separable Hilbert space H. However, in somecases, for example when one is interested in the excitation spectrum and dynamics of perturbationswith respect to a thermal equilibrium state of the system, the generator of the dynamics is notsemi-bounded and the Hilbert space may be non-separable. Therefore, in general, we will notassume that H is separable or that the Hamiltonian is bounded below.

As described in the introduction, we consider finite and infinite lattice systems with interactionsthat are sufficiently local. We allow for an infinite-dimensional Hilbert space at each site of thelattice. However, we impose conditions on the interactions that permit us to prove quasi-localitybounds of Lieb-Robinson type (in terms of the operator norm) for bounded local observables.This means that we will allow for the possibility of unbounded ‘spins’ and unbounded single-siteHamiltonians, but require that the interaction be given by bounded self-adjoint operators thatsatisfy a suitable decay condition at large distances (see Section 3 for more details). We do notconsider lattice oscillator systems with harmonic interactions in this paper, since one should notexpect bounds in terms of the operator norm for this class of systems (see [4, 89]). An interestingmodel that does fit in the framework presented here is the so-called quantum rotor model, whichhas an unbounded Hamiltonian for the quantum rotor at each site, but the interactions betweenrotors are described by a bounded potential [72, 77,115].

We will use the so-called interaction picture to describe the dynamics of Hamiltonians withunbounded on-site terms. This requires that we also consider time-dependent interactions. Time-dependent Hamiltonians are, of course, of interest in their own right, for instance in applicationsof quantum information theory. Therefore, we begin with a discussion of the Schrodinger equationfor the class of time-dependent Hamiltonians considered in this work.

Let H be a complex Hilbert space and B(H) denote the bounded linear operators on H. LetI ⊆ R be a finite or infinite interval. In this section we review some basic properties of the dynamicsof a quantum system with a time-dependent Hamiltonian of the form

(2.1) H(t)ψ = H0ψ +Φ(t)ψ, ψ ∈ D

where H0 is a time-independent self-adjoint operator with dense domain D ⊂ H, and for t ∈ I,Φ(t)∗ = Φ(t) ∈ B(H) and t 7→ Φ(t) is continuous in the strong operator topology. This meansthat for all ψ ∈ H, the function t 7→ Φ(t)ψ is continuous in the Hilbert space norm. From theseassumptions, it follows that for all t ∈ I, H(t) is self-adjoint with time-independent dense domainD, see [126][Theorem 5.28].

We will often consider operator-valued and vector-valued functions of one or more real (or com-plex) variables and impose various continuity assumptions, which we now briefly review. Anoperator-valued function is said to be norm continuous (norm differentiable) if it is continuous(differentiable) in the operator norm, and strongly continuous (strongly differentiable) if it is con-tinuous (differentiable) in the strong operator topology. With a slight abuse of terminology, wewill refer to Hilbert space-valued functions as strongly continuous (strongly differentiable) if theyare continuous (differentiable) in the Hilbert space norm. For transparency, when we considermaps defined on a linear space of operators, we will indicate the relevant topology and continuityassumptions explicitly.


The dynamics of a system described in (2.1) is determined by the Schrodinger equation:

(2.2)d

dtψ(t) = −iH(t)ψ(t), ψ(t0) = ψ0 ∈ D, t0 ∈ I.

For boundedH(t), through a standard construction, we will see that there exists a family of unitariesU(t, s) ∈ B(H), s, t ∈ I, that is jointly strongly continuous with ψ(t) = U(t, t0)ψ0 being the uniquesolution of (2.2) for all ψ0 ∈ H. It follows that the family U(t, s) has the co-cycle property: forr ≤ s ≤ t ∈ I, U(t, r) = U(t, s)U(s, r) and U(t, t) = 1. In the case that H(t) = H0 + Φ(t) whereH0 is an arbitrary unbounded self-adjoint operator and Φ(t) is bounded, we will make use of thewell-known interaction picture dynamics to construct an analogous unitary co-cycle. This co-cyclewill, in particular, generate the unique weak solution of the Schrodinger equation. To this end, wefirst discuss some other aspects of strongly continuous operator-valued functions that we will need.

2.1. Properties of continuity, measurability, and integration in B(H). In this section wereview some terminology and discuss a number of properties of operator-valued functions that willbe used extensively in the rest of the paper.

Let I ⊂ R be a finite or infinite interval and A : I → B(H) be strongly continuous, i.e., forall ψ ∈ H, t 7→ A(t)ψ is continuous with respect to the Hilbert space norm. By the UniformBoundedness Principle, if A is strongly continuous, then A is locally bounded, meaning if J ⊂ I iscompact, then

(2.3) MJ := supt∈J

‖A(t)‖ <∞.

The strong continuity of t 7→ A(t) implies that t 7→ ‖A(t)ψ‖ is continuous for all ψ ∈ H, and bythe above, the map t 7→ ‖A(t)‖ is locally bounded. However, strong continuity does not imply thatt 7→ ‖A(t)‖ is continuous, see [95, Section 2] for a counterexample.

We note that in this paper we use the notations ‖A(t)‖ and ‖A‖(t) interchangeably. For ease oflater reference, we now state a simple proposition.

Proposition 2.1. Let I ⊂ R be a finite or infinite interval and H and K be Hilbert spaces.

(i) If A,B : I → B(H) are strongly continuous (strongly differentiable), then (t, s) 7→ A(t)B(s)is jointly strongly continuous (separately strongly differentiable).

(ii) If A : I → B(H) and B : I → B(K) are strongly continuous (strongly differentiable), then(t, s) 7→ A(t)⊗B(s) is jointly strongly continuous (separately strongly differentiable).

(iii) If A : I → B(H) is strongly continuous, then the function t 7→ ‖A(t)‖ is lower semicontinu-ous, measurable, locally bounded and, hence, locally integrable.

It is clear that an analogue of Proposition 2.1 holds when strongly is replace with norm in thestatements above. Moreover, an argument similar to the one found in Proposition 2.1 (i) belowshows that if A : I → B(H) and ψ : I → H are both strongly continuous (strongly differentiable),then (t, s) 7→ A(t)ψ(s) is jointly strongly continuous (separately strongly differentiable). As will beclear from the proof, we note that the conclusions of part (iii) of this proposition continue to holdeven for weakly continuous A(t).

Proof. We prove the statements above in the case of strong continuity; the strong differentiabilityclaims follow similarly.

For (i), let ψ ∈ H and t0, s0 ∈ I be fixed. Given that A and B are strongly continuous, andhence locally bounded, it follows that A(t)B(s)ψ → A(t0)B(s0)ψ as (t, s) → (t0, s0) since

‖A(t)B(s)ψ −A(t0)B(s0)ψ‖ ≤ ‖A(t)‖‖(B(s) −B(s0))ψ‖+ ‖(A(t) −A(t0))B(s0)ψ‖.To prove (ii), we first show that if t 7→ A(t) ∈ B(H) is strongly continuous and K is another

Hilbert space, then the map t 7→ A(t)⊗1 ∈ B(H⊗K) is also strongly continuous. To see this, notethat for all ψ ∈ H ⊗ K, there exist two countable sets of vectors ψn ∈ H and φn ∈ K with φnn


orthonormal such that ψ =∑

n ψn ⊗ φn, and∑

n ‖ψn‖2 = ‖ψ‖2. Fix s ∈ I, and let J ⊆ I be acompact interval that contains a neighborhood of s. Using the orthonormality of φn, we find thatfor all t ∈ J

‖((A(t) −A(s))⊗ 1l)ψ‖2 =∑

n

‖(A(t)−A(s))ψn‖2.

For any ǫ > 0, choose N large enough so that∑

n>N ‖ψn‖2 ≤ ǫ/8M2J , where MJ > 0 satisfies (2.3).

By the strong continuity of A(t), there exists a δ > 0 such that for all t ∈ (s − δ, s + δ) ⊆ J and1 ≤ n ≤ N , one has ‖(A(t)−A(s))ψn‖2 < ǫ/2N. Putting these together, if |t− s| < δ, then

(2.4) ‖((A(t) −A(s))⊗ 1l)ψ‖2 ≤N∑

n=1

‖(A(t) −A(s))ψn‖2 +∑

n>N

(‖A(t)‖ + ‖A(s)‖)2‖ψn‖2 < ǫ.

Since A(t) ⊗ B(s) = (A(t) ⊗ 1l)(1l ⊗ B(s)), by (i), the tensor product of two strongly continuousmaps t 7→ A(t) ∈ B(H) and s 7→ B(s) ∈ B(K) is jointly strongly continuous.

To prove (iii), we start by noting that, by virtue of the strong continuity of A(t), the function‖A(t)‖ can be expressed as a supremum of continuous functions:

‖A(t)‖ = sup〈φ,A(t)ψ〉 | φ,ψ ∈ H, ‖φ‖ = ‖ψ‖ = 1.Recall that a function f : I → R is lower semicontinuous iff for all s ∈ R, f−1((s,∞)) is an opensubset of I. Now, if f is the supremum of a family of functions fα, we have that f−1((s,∞)) =⋃

α f−1α ((s,∞)). In our case, the fα are indexed by a pair of unit vectors in H and each fα is

continuous. Therefore, f−1α ((s,∞)) is open for all α and so is

⋃

α f−1α ((s,∞)). This shows the

lower semicontinuity.Since we have that f−1((s,∞)) is open, this set is also Borel measurable, for all s ∈ R. By a

standard lemma in measure theory [41], this implies that f is measurable.We already noted above that ‖A(t)‖ is bounded on compact intervals by the Uniform Bounded-

ness Principle. This concludes the proof.

We will make frequent use of integrals of vector-valued and operator-valued functions. It isstraightforward to define such integrals in the weak sense. In fact, if A : I → B(H) is locallybounded and weakly measurable, i.e. for all φ,ψ ∈ H, t 7→ 〈φ,A(t)ψ〉 is measurable, then for anycompact J ⊂ I the integral of A over J is defined as the operator BJ ∈ B(H) corresponding to thebounded sesquilinear form

(2.5) 〈φ,BJψ〉 =∫

J〈φ,A(t)ψ〉 dt .

We routinely use the notation BJ =∫

J A(t) dt to denote this operator. For strongly continuousfunctions A(t), the same integral can be interpreted in the strong sense:

(2.6)

(∫

JA(t) dt

)

ψ =

∫

JA(t)ψ dt,

where the RHS is understood to be the Riemann integral of a strongly continuous, Hilbert space-valued function. Since the range of any strongly continuous, Hilbert space-valued function belongsto a separable subspace (even if H is not separable, see, e.g., [129, Section V.4]), this integral alsoexists in the sense of Bochner.

The following well-known inequalities hold for all A : I → B(H), strongly continuous and J ⊂ Icompact:

(2.7) |〈φ,BJψ〉| ≤∫

J|〈φ,A(t)ψ〉|dt ≤ ‖φ‖

∫

J‖A(t)ψ‖dt,


and thus

(2.8)

∥

∥

∥

∥

(∫

JA(t)dt

)

ψ

∥

∥

∥

∥

≤∫

J‖A(t)ψ‖dt.

In particular, we obtain

(2.9)

∥

∥

∥

∥

∫

JA(t)dt

∥

∥

∥

∥

≤∫

J‖A(t)‖dt ≤ |J | sup

t∈J‖A(t)‖.

The first inequality extends to infinite J if, e.g., A(t) = B(t)w(t), with B(t) strongly continuousand bounded and w ∈ L1(J). Finally, it is easy to see that if A is strongly continuous, then

B(t) =∫ tt0A(s)ds is norm continuous, and strongly differentiable with d

dtB(t) = A(t). As such, thefundamental theorem of calculus also holds in the strong sense.

2.2. Dynamical equations and the Dyson series. In this section, we review some well-knownfacts about Dyson series and from them obtain the Schrodinger dynamics generated by a bounded,time-dependent Hamiltonian. A standard result in this direction can be summarized as follows.Let H be a Hilbert space, I ⊂ R be a finite or infinite interval, and H : I → B(H) be stronglycontinuous and pointwise self-adjoint, i.e. H(t)∗ = H(t) for all t ∈ I. Under these assumptions (see,e.g. Theorem X.69 of [109]) for each t0 ∈ I and every initial condition ψ0 ∈ H, the time-dependentSchrodinger equation

(2.10) id

dtψ(t) = H(t)ψ(t), ψ(t0) = ψ0,

has a unique solution in the sense that there is a unique, strongly differentiable function ψ : I → Hwhich satisfies (2.10). This solution can be characterized in terms of a two-parameter family ofunitaries U(t, s)s,t∈I ⊂ B(H) such that

(2.11) ψ(t) = U(t, s)ψ(s) for all s, t ∈ I .

These unitaries are often referred to as propagators, and an explicit construction of them is givenby the Dyson series. Specifically, for any s, t ∈ I and each ψ ∈ H the Hilbert space-valued series

(2.12) U(t, s)ψ = ψ +

∞∑

n=1

(−i)n∫ t

s

∫ t1

s· · ·∫ tn−1

sH(t1) · · ·H(tn)ψ dtn · · · dt1

is easily seen to be absolutely convergent in norm. One checks that U(t, s), as defined in (2.12),satisfies the differential equation

(2.13)d

dtU(t, s) = −iH(t)U(t, s), U(s, s) = 1l

which is to be understood in the sense of strong derivatives. Of course, under the stronger assump-tion that H : I → B(H) is norm continuous, then (2.13) also holds in norm.

The additional observation we want to make here is that U(t, s) is not only the unique strongsolution of (2.13); it is also the case that any bounded weak solution of (2.13) necessarily coincideswith U(t, s). By weak solution, we mean that for all φ,ψ ∈ H and any s, t ∈ I, U(t, s) satisfies

(2.14)d

dt〈φ,U(t, s)ψ〉 = −i〈φ,H(t)U(t, s)ψ〉, U(s, s) = 1l .

A proof of this fact is contained in the following proposition.

Proposition 2.2. Let A : I → B(H) be strongly continuous, and consider the differential equation

(2.15)d

dtV (t) = A(t)V (t), V (t0) = V0 ∈ B(H), t0 ∈ I.

The following statements hold:(i) There is a unique strong solution V : I → B(H) of (2.15), and V is norm continuous.


(ii) Any locally norm-bounded, weak solution of (2.15) coincides with the strong solution.(iii) Let D ⊂ H be dense. Suppose V : I → B(H) is strongly continuous and satisfies

(2.16)d

dtV (t)ψ = A(t)V (t)ψ, V (t0)ψ = V0ψ

for all ψ ∈ D and t ∈ I. Then, V is the unique strong solution.(iv) If V0 is invertible, the strong solution V of (2.15) is invertible for all t ∈ I. Moreover, in

this case, the inverse of V is the unique strong solution of

(2.17)d

dtV −1(t) = −V −1(t)A(t), V −1(t0) = V −1

0 ∈ B(H).

(v) If H : I → B(H) is strongly continuous and pointwise self-adjoint, then the strong solutionV of (2.15) with the choice A = −iH and V0 = 1l is unitary for all t ∈ I. Moreover, the mapU : I × I → B(H) given by U(t, s) = V (t)V (s)∗ is the unique strong solution to (2.13).

Before giving the proof of this proposition, we first comment on the content of part (v) andits relationship to the Dyson series from (2.12). A simple consequence of (i) is that the mappingU(t, s), defined in (v), is jointly norm continuous. As stated in (v), this U(t, s) is the unique strongsolution of (2.13); it is also strongly differentiable in s, and by (iv), this strong derivative is

(2.18)d

dsU(t, s) = iU(t, s)H(s) .

In addition, one readily checks that U(t, s)−1 = V (s)V (t)∗, for all s, t ∈ I, and thus U(t, s)−1 =U(t, s)∗ = U(s, t), for all s, t ∈ I, so that U(t, s) is a two-parameter family of unitaries. This familyof unitaries satisfies the co-cycle property: if r ≤ s ≤ t, then

(2.19) U(t, s)U(s, r) = U(t, r) and U(s, s) = 1l.

Finally, arguing as in the proof of (i) below, one sees that the Dyson series (2.12) is a strong solutionof (2.13). Combining this with the uniqueness proven in (v), we conclude that U(t, s) = V (t)V (s)∗

must coincide with the Dyson series constructed in (2.12).

Proof. (i) Define a map V : I → B(H) via a Dyson series, i.e. for any ψ ∈ H and each t ∈ I, set

(2.20) V (t)ψ = V0ψ +

∞∑

n=1

∫ t

t0

∫ t1

t0

· · ·∫ tn−1

t0

A(t1) · · ·A(tn)V0ψ dtn · · · dt1 .

We now argue that V is the unique strong solution of (2.15).First, we show that V is well-defined. The integrals appearing as terms in this series are well-

defined due to the strong continuity of A. More precisely, for any n ≥ 1, the product A(t1) · · ·A(tn)is jointly strongly continuous in the variables t1, · · · , tn, and thus the integrands are locally inte-grable. Next, for any t ≥ t0, the bound

‖V (t)ψ‖ ≤ ‖V0‖‖ψ‖ + ‖V0‖‖ψ‖∞∑

n=1

∫ t

t0

∫ t1

t0

· · ·∫ tn−1

t0

‖A‖(t1) · · · ‖A‖(tn) dtn · · · dt1

≤ ‖V0‖‖ψ‖∞∑

n=0

1

n!

(∫ t

t0

‖A‖(s) ds)n

(2.21)

holds. Here, we note that we are using the alternate notation ‖A‖(t) for ‖A(t)‖. As it is clear thata similar argument holds for t < t0, we see that V is well-defined as an absolutely convergent (innorm) series.


Next, we prove that V is a strong solution of (2.15). To see this, define recursively a sequence ofoperators Vnn≥1, Vn : I → B(H) by setting

(2.22) V1(t)ψ = A(t)V0ψ and Vn(t)ψ = A(t)

∫ t

t0

Vn−1(t1)ψ dt1 for any n ≥ 2 and t ∈ I .

With respect to this notation, it is clear that

(2.23) V (t)ψ = V0ψ +

∞∑

n=1

∫ t

t0

Vn(t1)ψ dt1

and moreover, for any h 6= 0,

V (t+ h)− V (t)

hψ =

∞∑

n=1

1

h

∫ t+h

tVn(t1)ψ dt1

=∞∑

n=1

Vn(t)ψ +∞∑

n=1

1

h

∫ t+h

t(Vn(t1)− Vn(t))ψ dt1 .(2.24)

Using the recursive definition, i.e. (2.22), it is clear that the first term on the right-hand-side aboveis A(t)V (t)ψ. A dominated convergence argument, using an estimate like (2.21), guarantees thatthe remainder term goes to zero in norm, and hence V is a strong solution.

Finally, we prove uniqueness. Let V1 and V2 be two strong solutions of (2.15). For any t ∈ I, sett+ = maxt, t0 and t− = mint, t0. Given ψ ∈ H, we have that

‖(V1(t)− V2(t))ψ‖ =

∥

∥

∥

∥

∫ t

t0

d

ds(V1(s)− V2(s))ψ ds

∥

∥

∥

∥

=

∥

∥

∥

∥

∫ t

t0

A(s)(V1(s)− V2(s))ψ ds

∥

∥

∥

∥

≤∫ t+

t−

‖A‖(s)‖(V1(s)− V2(s))ψ‖ ds

and uniqueness follows from Gronwall’s Lemma.One also sees that this V is norm continuous. In fact, let t, t0 ∈ I and ψ ∈ H. Clearly

(2.25) ‖(V (t)− V (t0))ψ‖ =

∥

∥

∥

∥

∫ t

t0

d

dsV (s)ψ ds

∥

∥

∥

∥

≤ ‖ψ‖∫ t+

t−

‖AV ‖(s) ds

and norm continuity of V follows.(ii) The uniqueness statement in (ii) is proven similarly. In fact, let V1 and V2 be two locally

norm-bounded weak solutions of (2.15). In this case, for any φ,ψ ∈ H and each t ∈ I,

|〈φ, (V1(t)− V2(t))ψ〉| =

∣

∣

∣

∣

∫ t

t0

d

ds〈φ, (V1(s)− V2(s))ψ〉 ds

∣

∣

∣

∣

=

∣

∣

∣

∣

∫ t

t0

〈φ,A(s)(V1(s)− V2(s))ψ〉 ds∣

∣

∣

∣

≤ ‖φ‖‖ψ‖∫ t+

t−

‖A‖(s)‖V1 − V2‖(s) ds.(2.26)

where we have used the notation t± as above. Taking the supremum of (2.26) over all normalizedφ,ψ ∈ H gives

(2.27) ‖V1 − V2‖(t) ≤∫ t+

t−

‖A‖(s)‖V1 − V2‖(s) ds .


Again, uniqueness follows from Gronwall’s Lemma. Since any strong solution is also a locallybounded weak solution, the unique weak and unique strong solutions must coincide.

(iii) We will show that any V satisfying the assumptions of (iii) is actually the locally boundedweak solution. To this end, note that for any φ ∈ H, (2.16) implies

(2.28)d

dt〈φ, V (t)ψ〉 = 〈φ,A(t)V (t)ψ〉

holds for all ψ ∈ D and any t ∈ I. Let ψ ∈ H and take any sequence ψnn≥1 in D with ψn

converging to ψ. Consider the sequence of functions fn : I → C defined by

(2.29) fn(t) = 〈φ, V (t)ψn〉 for all t ∈ I,

and set f(t) = limn→∞ fn(t). Note that these pointwise limits exist as ψn converges to ψ and Vis locally bounded. One also sees that f(t) = 〈φ, V (t)ψ〉 for all t ∈ I. From (2.28), it is clearthat f ′n(t) = 〈φ,A(t)V (t)ψn〉. Since AV is strongly continuous, and hence locally bounded, thesame argument shows that g : I → C with g(t) = limn→∞ f ′n(t) = 〈φ,A(t)V (t)ψ〉 is well-defined.Observing further that f ′n converges to g uniformly on compact subsets of I, it is clear that theconditions of [113, Theorem 7.17] are satisfied. We conclude that f ′(t) = g(t) for all t ∈ I andhence, V is the unique locally bounded weak solution. By the result proven in (ii), V also coincideswith the unique strong solution.

(iv) Arguing as in the proof of (i), the function W : I → B(H) defined by setting

(2.30) W (t)ψ = V −10 ψ +

∞∑

n=1

(−1)n∫ t

t0

∫ t1

t0

· · ·∫ tn−1

t0

V −10 A(tn) · · ·A(t1)ψ dtn · · · dt1

for any ψ ∈ H is a strong solution of the initial value problem

(2.31)d

dtW (t) = −W (t)A(t), W (t0) = V −1

0 .

Now, with V the strong solution of (2.15), consider the function Y : I → B(H) given by Y (t) =V (t)W (t). One checks that

(2.32)d

dtY (t) = A(t)Y (t)− Y (t)A(t), Y (t0) = 1,

holds in the strong sense. It is clear that Y (t) = 1 solves the above initial value problem. AGronwall argument, similar to those we have proven before, shows that this constant solution isthe unique strong solution of (2.32) and thus, W is a right inverse of V for all t ∈ I. Noting thatthe function Z : I → B(H) defined by Z(t) =W (t)V (t) satisfies the trivial initial value problem:

(2.33)d

dtZ(t) = 0, Z(t0) = 1,

we conclude W (t) = V (t)−1 as claimed. In fact, uniqueness of the strong solution of (2.31) follows.(v) One sees that V is unitary by noting that the adjoint of the operator defined by (2.20) agrees

with the Dyson series for V −1 given in (2.30). For each s ∈ I fixed, the map t 7→ V (t)V (s)∗ definesa strong solution of (2.13) and by (i) it is unique.

We conclude this section with an estimate on the solution of certain dynamical equations thatwill be useful in the proof of the Lieb-Robinson bound in Section 3.

Lemma 2.3. Let H be a Hilbert space, I ⊂ R be a finite or infinite interval, and A,B : I → B(H)be strongly continuous with A pointwise self-adjoint, i.e. A(t)∗ = A(t) for all t ∈ I. For each t0 ∈ Iand V0 ∈ B(H), the initial value problem

(2.34)d

dtV (t) = −i[A(t), V (t)] +B(t) with V (t0) = V0


has a unique strong solution. In particular,

(2.35) ‖V ‖(t) ≤ ‖V0‖+∫ t+

t−

‖B‖(s) ds

where t+ = maxt, t0, t− = mint, t0. Moreover, any locally bounded weak solution of (2.34)coincides with the strong solution and, therefore, satisfies the estimate in (2.35).

Proof. By Proposition 2.2 (v), the unique strong solution of

(2.36)d

dtW (t) = −iA(t)W (t) with W (t0) = 1l

is unitary for all t ∈ I. As a product of strongly differentiable maps, V : I → B(H) given by

(2.37) V (t) =W (t)

(

V0 +

∫ t

t0

W (s)∗B(s)W (s) ds

)

W (t)∗

is strongly differentiable. In fact, a short calculation shows that this V is a strong solution of (2.34),and moreover, the bound claimed in (2.35) is clear. Arguments involving Gronwall’s lemma, similarto those found in the proof of Proposition 2.2 (i) and (ii), verify the claimed uniqueness results.

2.3. The dynamics for a class of unbounded Hamiltonians.

2.3.1. On the interaction picture dynamics. The following proposition is an important applicationof Proposition 2.2. As explained in the remarks of Section X.12 of [109], applying the interactionpicture representation to Hamiltonians with the form H = H0 +Φ, even if Φ is time-independent,leads one to study a dynamics with time-dependent Hamiltonians. In this situation, one oftenproduces Hamiltonians that are strongly continuous, but not norm continuous. This leads us toconsider Hamiltonians of the form H(t) = H0+Φ(t), where H0 is a self-adjoint operator with densedomain D, and Φ(t) is a bounded, pointwise self-adjoint operator that is strongly continuous in t.

Proposition 2.4. Let H be a Hilbert space and H0 a self-adjoint operator with dense domainD ⊂ H. Let I ⊂ R be a finite or infinite interval and Φ : I → B(H) be strongly continuous andpointwise self-adjoint. Then, there is a two parameter family of unitaries U(t, s)s,t∈I associatedto the self-adjoint operator H(t) = H0 +Φ(t) for which:

(i) (t, s) 7→ U(t, s) is jointly strongly continuous,(ii) U(t, s) satisfies the co-cycle property (2.19),(iii) U(t, s) generates the unique, locally bounded weak solutions of the Schrodinger equation

associated to H(t), i.e. for any t0 ∈ I and ψ0 ∈ H, ψ : I → H given by ψ(t) = U(t, t0)ψ0

satisfies

(2.38)d

dt〈φ,ψ(t)〉 = −i〈H(t)φ,ψ(t)〉 with ψ(t0) = ψ0

for all φ ∈ D and t ∈ I.Proof. Since H0 is self-adjoint, Stone’s theorem implies that eitH0t∈R is a strongly continuous,

one-parameter unitary group. In this case, the map H : I → B(H) given by

(2.39) H(t) = eitH0Φ(t)e−itH0

is clearly pointwise self-adjoint and strongly continuous. Using Proposition 2.2 (v), we concludethat the unique strong solutions of

(2.40)d

dtU(t, s) = −iH(t)U (t, s) with U(s, s) = 1l


form a two-parameter family of unitaries U(t, s)s,t∈I which satisfy the co-cycle property (2.19).In terms of this family, we define U : I × I → B(H) by setting

(2.41) U(t, s) = e−itH0U(t, s)eisH0 .

One checks that U(t, s)s,t∈I is a two-parameter family of unitaries satisfying (i) and (ii) above.To prove (iii), let t0 ∈ I and ψ0 ∈ H. Define ψ : I → H by setting ψ(t) = U(t, t0)ψ0. Observe

that for any φ ∈ D and each t ∈ I

(2.42) 〈φ,ψ(t)〉 = 〈eitH0φ, U(t, t0)eit0H0ψ0〉,

with the right-hand-side being a differentiable function of t. One calculates that

d

dt〈φ,ψ(t)〉 = 〈iH0e

itH0φ, U(t, t0)eit0H0ψ0〉+ 〈eitH0φ,

d

dtU(t, t0)e

it0H0ψ0〉

= 〈iH0φ,ψ(t)〉 + 〈eitH0φ,−ieitH0Φ(t)ψ(t)〉= −i〈H(t)φ,ψ(t)〉(2.43)

as claimed.We need only justify uniqueness of the locally bounded weak solutions. Let t0 ∈ I, ψ0 ∈ H, and

suppose ψ1 and ψ2 are two locally bounded solutions of the initial value problem (2.38). Consider

the functions ψ1(t) = eitH0ψ1(t) and ψ2(t) = eitH0ψ2(t). It is easy to check that these functions arelocally bounded weak solutions of the Schrodinger equation associated to the bounded HamiltonianH(t) in (2.39). As such, they are unique, which may be argued as in the proof of Proposition 2.2,and therefore, so too are ψ1 and ψ2.

In this work, we define the Heisenberg dynamics on a suitable algebra of observables in termsof the strongly continuous propagator U(t, s) whose existence is guaranteed by Proposition 2.4.We work under assumptions that guarantee the uniqueness of bounded weak solutions. Strictlyspeaking, the uniqueness of the weak solution and the possible absence of a strong solution to theSchrodinger equation in the Hilbert space will play no role in our analysis. More information aboutthe solutions and their uniqueness could, however, be important for the unambiguous interpreta-tion of our results. Additional results exist in the literature if one is willing to make additionalassumptions on H0 and Φ(t). For example, the following theorem establishes the existence of aninvariant domain for the generator and, consequently, the existence of a unique strong solution forthe situation where H0 is semi-bounded and Φ(t) is Lipschitz continuous, which is a common phys-ical situation. As explained in the introduction, there are important applications of the methodsin this paper to situations where these additional assumptions are not satisfied.

Theorem 2.5. Let H0 be a self-adjoint operator with dense domain D ⊂ H and suppose H0 ≥ 0.Suppose Φ : R → B(H) is pointwise self-adjoint and ‘Lipschitz’ continuous in the sense that forany bounded interval I ⊂ R, there exists a constant C such that for all s, t ∈ I, we have

(2.44) ‖(H0 + 1)−1(Φ(t)− Φ(s))(H0 + 1)−1‖ ≤ C|t− s|.Then, there exists a strongly continuous propagator U(t, s), such that U(t, s)D ⊂ D, for all s ≤ t ∈I, and such that t 7→ U(t, t0)ψ0 is the unique strong solution of

(2.45)d

dtψ(t) = −i(H0 +Φ(t))ψ(t) with ψ(t0) = ψ0

for all ψ0 ∈ D.

In [119, Theorem II.21] Simon credits a version of this theorem to Yosida, who proved it in amore general Banach space context [129, Section XIV.4], but with the Lipschitz condition replacedby a boundedness condition on the derivative of Φ(t). Yosida gives credit to Kato [65, 66] andKisynski [70].


2.3.2. A Duhamel formula for bounded perturbations depending on a parameter. In this section, weconsider families of Hamiltonians Hλ(t) which depend on a time-parameter t ∈ I and an auxillaryparameter λ ∈ J . For such families, we will prove a version of the well-known Duhamel formula(Proposition 2.6) and use it to derive various continuity properties of the corresponding dynamics(Proposition 2.7).

Let H0 be a densely defined, self-adjoint operator on a Hilbert space H and denote by D ⊂ Hthe corresponding dense domain. Let I, J ⊂ R be intervals and consider the family of HamiltoniansHλ(t), t ∈ I and λ ∈ J , acting on D ⊂ H given by

(2.46) Hλ(t) = H0 +Φλ(t)

where for each t ∈ I and λ ∈ J , Φλ(t)∗ = Φλ(t) ∈ B(H). The self-adjointness of Hλ(t) on the

common domain, D, is clear. We will assume that (t, λ) 7→ Φλ(t) is jointly strongly continuous.We will also assume that for each fixed t ∈ I, the mapping λ 7→ Φλ(t) is strongly differentiable andthat the corresponding derivative, which we denote by Φ′

λ(t), satisfies that the map (t, λ) 7→ Φ′λ(t)

is jointly strongly continuous.Under these assumptions, Proposition 2.4 guarantees that for each λ ∈ J there exists a two

parameter family of unitaries Uλ(t, s)s,t∈I which generates the weak solutions of the Schrodingerequation associated to Hλ(t), see (2.38). Our goal here is to show that for fixed s, t ∈ I, the mapλ 7→ Uλ(t, s) is strongly differentiable and moreover,

(2.47)

∥

∥

∥

∥

d

dλUλ(t, s)

∥

∥

∥

∥

≤∫ max(s,t)

min(s,t)‖Φ′

λ‖(r)dr.

We will obtain this bound as a corollary of the following proposition, which gives a Duhamel formulafor the derivative in this setting. Although the Duhamel formula is well-known, we give an explicitproof here that allows us to clarify the continuity properties implied by our assumptions. In theproof we avoid taking derivatives with respect to t or s which, in general, are unbounded operators.

Proposition 2.6 (Duhamel Formula). Let Hλ(t) be a family of self-adjoint operators as in (2.46)above and let Uλ(t, s) denote the corresponding unitary propagator. Then, for all s, t ∈ I with s ≤ t,we have that

(2.48)d

dλUλ(t, s) = −i

∫ t

sUλ(t, r) Φ

′λ(r)Uλ(r, s) dr

where the derivative and the integral are to be understood in the strong sense.

With stronger assumptions, one can prove (2.48) holds in norm. In fact, arguing as below, if

(i) the map (t, λ) 7→ Φλ(t) is jointly norm continuous,(ii) for each t ∈ I, the map λ 7→ Φλ(t) is norm differentiable; with derivative denoted by Φ′

λ(t),and

(iii) the map (t, λ) 7→ Φ′λ(t) is jointly norm continuous,

then λ 7→ Uλ(t, s) is norm differentiable and its derivative satisfies (2.48).

Proof. Recall that the unitary propagator Uλ(t, s), as defined in the proof of Proposition 2.4, is

(2.49) Uλ(t, s) = e−itH0Uλ(t, s)eisH0

where Uλ(t, s) is the unique strong solution of

(2.50)d

dtUλ(t, s) = −iΦλ(t)Uλ(t, s) with Uλ(s, s) = 1l and Φλ(t) = eitH0Φλ(t)e

−itH0 .

We first prove the analogue of (2.48) for Uλ(t, s), i.e.

(2.51)d

dλUλ(t, s) = −i

∫ t

sUλ(t, r)Φ

′λ(r)Uλ(r, s) dr.


Given (2.51), the λ-derivative of (2.49) is easily seen to satisfy (2.48).We now show (2.51). The unique strong solution of (2.50) is given by the Dyson series

(2.52) Uλ(t, s) = 1+

∞∑

n=1

(−i)n∫ t

s

∫ t1

s· · ·∫ tn−1

sΦλ(t1) · · · Φλ(tn) dtn · · · dt1 .

To each n ≥ 1, define a map Ψλ : In → B(H) by setting

(2.53) Ψλ(t1, · · · , tn) = Φλ(t1) · · · Φλ(tn) for any (t1, · · · .tn) ∈ In .

With (t1, · · · , tn) ∈ In fixed, our assumptions imply that λ 7→ Ψλ(t1, · · · , tn) is strongly differen-tiable and moreover

(2.54)d

dλΨλ(t1, · · · , tn) =

n∑

k=1

Φλ(t1) · · · Φλ(tk−1)Φ′λ(tk)Φλ(tk+1) · · · Φλ(tn).

The joint strong continuity of (t, λ) 7→ Φλ(t) and (t, λ) 7→ Φ′λ(t) can be used to justify term-by-term

differentiation of the Dyson series (2.52), and we obtain

(2.55)d

dλUλ(t, s) =

∞∑

n=1

(−i)n∫ t

s· · ·∫ tn−1

s

d

dλΨλ(t1, · · · , tn) dtn · · · dt1 .

The proof of (2.51) is now completed by demonstrating that upon inserting the Dyson series for

Uλ(t, r) and Uλ(r, s) into the integral on the right-hand-side of (2.51), the result simplifies to theexpression on the right-hand-side of (2.55).

Note that upon substitution of (2.52) into the right-hand-side of (2.51) we find

−i∫ t

sUλ(t, r)Φ

′λ(r)Uλ(r, s) dr =

∑

p,q≥0

(−i)p+q+1

∫ t

s

∫ t

r

∫ t1

r· · ·∫ tp−1

r

∫ r

s

∫ tp+2

s· · ·∫ tp+q

s

×Φλ(t1) · · · Φλ(tp)Φ′λ(r)Φλ(tp+2) · · · Φλ(tp+q+1)(2.56)

×dtp+q+1 · · · dtp+2dtp · · · dt1dr.Here p (respectively q) is the index of the terms in the series for the first (respectively second) prop-agator, and we have taken as integration variables t1, . . . , tp and tp+2, . . . , tp+q+1. Each integrandabove is the product of n = p + q + 1 ≥ 1 operators. Since the goal is to re-write the above as in(2.55), we now re-index by writing p = k− 1 and q = n− k for n ≥ 1 and 1 ≤ k ≤ n. One sees that

−i∫ t

sUλ(t, r)Φ

′λ(r)Uλ(r, s) dr =

∞∑

n=1

n∑

k=1

(−i)n∫ t

s

∫ t

r

∫ t1

r· · ·∫ tk−2

r

∫ r

s

∫ tk+1

s· · ·∫ tn−1

s

×Φλ(t1) · · · Φλ(tk−1)Φ′λ(r)Φλ(tk+1) · · · Φλ(tn)(2.57)

×dtn · · · dtk+1dtk−1 · · · dt1dr.The identity (2.51) now follows by comparing, term by term, the integration domains on the right-hand-sides of (2.55) and (2.57). That they are equal follows, e.g., by reordering the iterated integralsin (2.57).

For each λ ∈ J , the Heisenberg dynamics τλt,s, s, t ∈ I, associated to the family of Hamiltoniansin (2.46) is the co-cycle of automorphisms of B(H) given by

(2.58) τλt,s(A) = Uλ(t, s)∗AUλ(t, s) for all A ∈ B(H).

As it will be convenient for later applications, we summarize various continuity properties of thisdynamics in the following proposition.

Proposition 2.7. Let Hλ(t) be a family of Hamiltonians as described in (2.46). The correspondingdynamics, as in (2.58) above, has the following properties:


(i) For each λ ∈ J and A ∈ B(H), the map (s, t) 7→ τλt,s(A) is jointly strongly continuous.

(ii) For each s, t ∈ I and A ∈ B(H), the map λ 7→ τλt,s(A) is strongly differentiable (and hencestrongly continuous). Moreover, one has the estimate

(2.59)

∥

∥

∥

∥

d

dλτλt,s(A)

∥

∥

∥

∥

≤ 2‖A‖∫ max(s,t)

min(s,t)‖Φ′

λ‖(r)dr.

(iii) For fixed s, t ∈ I and λ ∈ J , the map τλt,s(·) : B(H) → B(H) is continuous on bounded setswhen both its domain and codomain are equipped with the strong operator topology. Thiscontinuity is uniform for λ in compact subsets of J .

Proof. The statement in (i) follows from Proposition 2.4 as τλt,s(A), see (2.58), is the product ofjointly strongly continuous mappings.

To prove (ii), we use Duhamel’s formula from Proposition 2.6 to calculate the derivative. Specif-ically, note that if s ≤ t, then

(2.60)d

dλτλt,s(A) = i

∫ t

sτλr,s([Φ

′λ(r), τ

λt,r(A)]) dr.

An estimate of the form in (2.59) is now clear.To prove (iii), fix s, t ∈ I, and let [a, b] ⊂ J . Without loss of generality, assume that s ≤ t. Let

ǫ > 0. Since (r, λ) 7→ Φ′λ(r) is jointly strongly continuous,

(2.61) M := sup(r,λ)∈[s,t]×[a,b]

‖Φ′λ(r)‖ <∞.

Take δ > 0 so that

(2.62) 2δ(t − s)M ≤ ǫ.

By compactness, there is some N ≥ 1 and numbers λ1, · · · , λN ∈ [a, b] for which the balls of radiusδ centered at λi, 1 ≤ i ≤ N , cover [a, b]. Using the result in (ii), we see that for every λ ∈ [a, b]there is some 1 ≤ i ≤ N for which

(2.63) ‖τλt,s(A) − τλit,s(A)‖ ≤ ǫ‖A‖ for all A ∈ B(H) .

Now, to prove the continuity statement claimed, let Ann≥1 ⊆ B(H) be a bounded sequence thatconverges to A ∈ B(H) in the strong operator topology. Let B <∞ be such that supn≥1 ‖An‖ ≤ B.Using (2.58) and the strong convergence of An to A, it is easy to verify that for any ψ ∈ H and

any 1 ≤ i ≤ N , the sequence τλit,s(An)ψn≥1 converges to τλi

t,s(A)ψ in H. Pick n0 ≥ 1 so that forall n ≥ n0 and each 1 ≤ i ≤ N ,

(2.64) ‖τλit,s(An)ψ − τλi

t,s(A)ψ‖ ≤ ǫB‖ψ‖.

In this case, for any λ ∈ [a, b] there is an i for which

‖τλt,s(An)ψ − τλt,s(A)ψ‖ ≤ ‖τλt,s(An)ψ − τλit,s(An)ψ‖

+‖τλit,s(An)ψ − τλi

t,s(A)ψ‖+‖τλi

t,s(A)ψ − τλt,s(A)ψ‖≤ 3ǫB‖ψ‖

whenever n ≥ n0. This proves that the strong convergence is uniform for λ ∈ [a, b], or in otherwords, that the family of maps τλt,s(·) | λ ∈ [a, b], for s, t ∈ I fixed, is equicontinuous on boundedsets in B(H) with respect to the strong operator topology.


3. Lieb-Robinson bounds and infinite volume dynamics of lattice systems

The scope of this paper is lattice models with possibly unbounded single-site Hamiltonians andbounded interactions that, in general, may be time-dependent. This is the setting in which oneexpects to obtain Lieb-Robinson bounds with estimates in terms of the operator norm of theobservables. A well-known example of this situation is the quantum rotor model. We will notconsider lattice models with unbounded interactions in this work. The only systems with unboundedinteractions that have been studied so far are oscillator lattice systems for which the interactionsare quadratic [34] or bounded perturbations of quadratic interactions [4, 89] .

In this paper, the ‘lattice’ in lattice systems is understood to be a countable metric space (Γ, d)(not necessarily a lattice in the sense of the linear combinations with integer coefficients of a set ofbasis vectors in Euclidean space). Typically, Γ is infinite (or more specifically, has infinite diameter),and models are given in terms of Hamiltonians for a family of finite subsets of Γ. After an initialanalysis of the finite systems, we study the thermodynamic limit through sequences of increasingand absorbing finite volumes Λn, i.e. Λn ↑ Γ. Often, the goal is to obtain estimates for the finitesystems defined on Λn that are uniform in n. The definitions below prepare for this goal. We notethat it is perfectly possible to consider a finite set Γ and apply the results derived in this and thesubsequent sections to finite systems. We note that some of the conditions we impose are triviallysatisfied for finite systems.

The points of Γ, also called sites of the lattice, label a family of ‘small’ systems, which are often,but not necessarily, identical copies of a given system such as a spin, a particle in a confiningpotential such as a harmonic oscillator, or a quantum rotor. The quantum many-body latticesystems of condensed matter physics are of this type. A wide range of interesting behaviors arisesdue to interactions between the component systems. It is a central feature of extended physicalsystems that interactions have a local structure, meaning that the strength of the interactionsdecreases with the distance between the systems. Often, each system only interacts directly withits nearest neighbors in the lattice. The mean-field approximation ignores the geometry of theambient space and it often is a good first approximation. In more realistic models, however, theinteractions between different components depends on the distance between them. In this sectionwe derive a fundamental property of the dynamics of quantum lattice systems that is intimatelyrelated to the local structure of the interactions. This property is referred to as quasi-locality andits basic feature is a bound on the speed of propagation of disturbances in the system, which isknown as a Lieb-Robinson bound.

Lieb and Robinson were the first to derive bounds of this type [75]. In the years followingthe original article, a number of further important results appeared, e.g., by Radin [107] and inparticular by Robinson [112] who gave a new proof of the theorem of Lieb and Robinson (whichis included in [20]). Robinson also showed that Lieb-Robinson bounds can be used to prove theexistence of the thermodynamic limit of the dynamics and used the bounds to derive fundamentallocality properties of quantum lattice systems. It was only much later however, that Hastingswho pointed out how the Lieb-Robinson bounds could be used to prove exponential clustering ingapped ground states in a paper where he provided the first generalization of the Lieb-Schultz-Mattis theorem to higher dimensions [54]. Mathematical proofs then followed by Nachtergaele andSims [92], Hastings and Koma [57] and Nachtergaele, Ogata, and Sims [88]. The new approach toproving Lieb-Robinson bounds developed in these works leading to [94], yields a better prefactorwith a more accurate dependence on the support of the observables. This was important for certainapplications such as the proof of the split property for gapped ground states in one dimension byMatsui [81,82].

Further extensions of the Lieb-Robinson bounds in several directions quickly followed: Lieb-Robinson bounds for lattice fermions [24, 57, 97], Lieb-Robinson bounds for irreversible quantum


dynamics [53,98,105], a bound for certain long-range interactions [45,111,124], anomalous or zero-velocity bounds for disordered and quasi-periodic systems [27, 35, 36, 52], propagation estimatefor lattice oscillator systems [4, 26, 34, 89] and other systems with unbounded interactions [106],including classical lattice systems [28,61,108].

The list of applications of Lieb-Robinson bounds includes a broad range of topics: Lieb-Schultz-Mattis theorems [54,93], the entanglement area law in one dimension [55], the quantum Hall effect[6,44,58], quasi-adiabatic evolution (spectral flow and automorphic equivalence) including stabilityand classification of gapped ground state phases [12, 21, 22, 59, 83, 96], the stability of dissipativesystems [19, 76], quasi-particle structure of the excitation spectrum of gapped systems [10, 50], astability property of the area law of entanglement [78], the efficiency of quantum thermodynamicengines [118], the adiabatic theorem and linear response theory for extended systems [7, 8], thedesign and analysis of quantum algorithms [49], and the list continues to grow: [5–7, 25, 30, 38, 47,64,84,123].

In order to express the locality properties of the interactions and the resulting dynamics, weintroduce some additional structure on the discrete metric space (Γ, d) in the next section.

3.1. Lieb-Robinson estimates for bounded time-dependent interactions.

3.1.1. General setup. As described above, we will study quantum lattice models with possiblyunbounded single-site Hamiltonians but bounded, in general, time-dependent interactions. In thissection, we give the framework for quantum lattice systems and describe the bounded interactionsof interest. We will consider the addition of unbounded on-site Hamiltonians in later sections.

The lattice models we consider are defined over a countable metric space (Γ, d). To each sitex ∈ Γ, we associate a complex Hilbert space Hx and denote the algebra of all bounded linearoperators on Hx by B(Hx). Let P0(Γ) be the collection of all finite subsets of Γ. For any Λ ∈ P0(Γ),the Hilbert space of states and algebra of local observables over Λ are denoted by

(3.1) HΛ :=⊗

x∈Λ

Hx and AΛ :=⊗

x∈Λ

B(Hx) = B(HΛ),

where we have chosen to define the tensor product of the algebras B(Hx) so that the last equalityholds (i.e. the spatial tensor product, corresponding to the minimal C∗-norm [116]). For any twofinite sets Λ0 ⊂ Λ ⊂ Γ, each A ∈ AΛ0 can be naturally identified with A ⊗ 1Λ\Λ0

∈ AΛ. Withrespect to this identification, the algebra of local observables is then defined as the inductive limit

(3.2) AlocΓ =

⋃

Λ∈P0(Γ)

AΛ,

and the C∗-algebra of quasi-local observables, which we denote by AΓ, is the completion of AlocΓ

with respect to the operator norm. We will use the phrase quantum lattice system to mean thecountable metric space (Γ, d) and quasi-local algebra AΓ.

A model on a quantum lattice system is given in terms of an interaction Φ. In the time-independent case, this is a map Φ : P0(Γ) → Aloc

Γ such that Φ(Z)∗ = Φ(Z) ∈ AZ for all Z ∈ P0(Γ).The quantum lattice model associated to Φ is the collection of all local Hamiltonians of the form

(3.3) HΛ =∑

X⊂Λ

Φ(X), Λ ∈ P0(Γ).

We will also consider time-dependent interactions. Let I ⊂ R be an interval. A map Φ :P0(Γ)× I → Aloc

Γ is said to be a strongly continuous interaction if:

(i) To each t ∈ I, the map Φ(·, t) : P0(Γ) → AlocΓ is an interaction.

(ii) For each Z ∈ P0(Γ), Φ(Z, ·) : I → AZ is strongly continuous.


Given such a strongly continuous interaction Φ, we will often denote by Φ(t) the interaction Φ(·, t)as in (i) above, and define the corresponding local Hamiltonians

(3.4) HΛ(t) =∑

Z⊂Λ

Φ(Z, t) for Λ ∈ P0(Γ).

Analogous to the above, a corresponding time-dependent quantum lattice model may be defined.By our assumptions on the interaction, it is clear that for each t ∈ I, HΛ(t) is a bounded, self-adjoint operator on HΛ. Moreover, by Proposition 2.1 HΛ : I → AΛ is strongly continuous.In this case, Proposition 2.2 demonstrates that there exists a two-parameter family of unitariesUΛ(t, s)s,t∈I ⊂ AΛ, defined as the unique strong solution of the initial value problem

(3.5)d

dtUΛ(t, s) = −iHΛ(t)UΛ(t, s), UΛ(s, s) = 1 , for all s, t ∈ I.

In terms of these unitary propagators, we define a Heisenberg dynamics τΛt,s : AΛ → AΛ by setting

(3.6) τΛt,s(A) = UΛ(t, s)∗AUΛ(t, s) for all A ∈ AΛ.

In some applications, including Theorem 3.1 below, we will also consider the inverse dynamics,

(3.7) τΛt,s(A) := UΛ(t, s)AUΛ(t, s)∗ = τΛs,t(A),

where the final equality follows from Proposition 2.2 (iv).As discussed above, Lieb-Robinson bounds approximate the speed of propagation of dynamically

evolved observables through a quantum lattice system, and this estimate is closely tied to thelocality of the interaction in question. To quantify the locality of an interaction, we introduce thenotion of an F -function. An F -function on (Γ, d) is a non-increasing function F : [0,∞) → (0,∞),satisfying the following two properties:

(i) F is uniformly integrable over Γ, i.e.

(3.8) ‖F‖ = supx∈Γ

∑

y∈Γ

F (d(x, y)) <∞,

(ii) F satisfies the convolution condition

(3.9) CF = supx,y∈Γ

∑

z∈Γ

F (d(x, z))F (d(z, y))

F (d(x, y))<∞.

An equivalent formulation of (ii) is that there exists a constant C <∞ such that

(3.10)∑

z∈Γ

F (d(x, z))F (d(z, y)) ≤ CFF (d(x, y)), for all x, y ∈ Γ.

Let F be an F -function on (Γ, d) and g : [0,∞) → [0,∞) be any non-decreasing, subadditivefunction, i.e. g(r + s) ≤ g(r) + g(s) for all r, s ∈ [0,∞). Then, the function

(3.11) Fg(r) = e−g(r)F (r),

also satisfies (i) and (ii) with ‖Fg‖ ≤ ‖F‖ and CFg ≤ CF . We call any F -function of this form aweighted F-function.

It is easy to produce examples of these F -functions when Γ = Zν for some ν ≥ 1 and d(x, y) =

|x− y| is the ℓ1-distance. In fact, for any ǫ > 0 the function

(3.12) F (r) =1

(1 + r)ν+ǫ

is an F -function on Zν . It is clear that this function is uniformly integrable, i.e. (3.8) holds.

Moreover, one may verify that

(3.13) CF ≤ 2ν+ǫ‖F‖ .


In the special case of g(r) = ar, for some a ≥ 0, we obtain a very useful family of weighted F -functions, which we denote by Fa, given by Fa(r) = e−ar/(1+r)ν+ǫ. See Appendix 8.1-8.3 for otherexamples and properties of F -functions.

We use these F -functions to describe the decay of a given interaction. Let F be an F -functionon (Γ, d) and Φ : P0(Γ) → Aloc

Γ be an interaction. The F -norm of Φ is defined by

(3.14) ‖Φ‖F = supx,y∈Γ

1

F (d(x, y))

∑

Z∈P0(Γ):

x,y∈Z

‖Φ(Z)‖.

It is clear from the above equation that for all x, y ∈ Γ,

(3.15)∑

Z∈P0(Γ):

x,y∈Z

‖Φ(Z)‖ ≤ ‖Φ‖FF (d(x, y)).

Note that for any Z ∈ P0(Γ), there exist x, y ∈ Z for which d(x, y) = diam(Z); the latter being thediameter of Z. In this case, a simple consequence of (3.15) is

(3.16) ‖Φ(Z)‖ ≤∑

Z′∈P0(Γ):

x,y∈Z′

‖Φ(Z ′)‖ ≤ ‖Φ‖FF (diam(Z)).

We will be mainly interested in situations where the quantity in (3.14) is finite. In this case, thebound (3.16) demonstrates that the F -function governs the decay of an individual interaction term,and moreover, the estimate (3.15) generalizes this notion of decay by including all interaction termscontaining a fixed pair of points x and y.

When Γ is finite, then ‖Φ‖F is finite for any interaction Φ and any function F . For infinite Γ,the set of interactions Φ for which ‖Φ‖F < ∞ depends on F . It is easy to check that ‖ · ‖F is anorm on the set of interactions for which it is finite. In terms of this norm, we define the Banachspace

(3.17) BF = Φ : P0(Γ) → AlocΓ | Φ is an interaction and ‖Φ‖F <∞.

Of course, BF depends on Γ and on the single-site Hilbert spaces Hx, but that information willalways be clear from the context.

We introduce an analogue of (3.14) for time-dependent interactions as follows. Consider a quan-tum lattice system comprised of (Γ, d) and AΓ. Let I ⊂ R be an interval and Φ : P0(Γ)× I → Aloc

Γbe a strongly continuous interaction. Given an F -function on (Γ, d), we will denote by BF (I) thecollection of all strongly continuous interactions Φ for which the mapping

(3.18) ‖Φ(t)‖F = supx,y∈Γ

1

F (d(x, y))

∑

Z∈P0(Γ):

x,y∈Z

‖Φ(Z, t)‖, for t ∈ I

is locally bounded. As with the operator norm, we will sometimes use the alternate notation‖Φ‖F (t) for the quantity defined in (3.18). The function t 7→ ‖Φ‖F (t) is measurable since it is thesupremum of a countable family of measurable functions. As such, ‖Φ‖F is locally integrable. Asin the time-independent case, (3.18) implies that for all t ∈ I and x, y ∈ Γ,

(3.19)∑

Z∈P0(Γ):

x,y∈Z

‖Φ(Z, t)‖ ≤ ‖Φ‖F (t)F (d(x, y)),

a bound which will appear in many of our estimates. See Appendix 8.4 for more useful estimatesinvolving interactions Φ ∈ BF (I).


3.1.2. Lieb-Robinson estimates for bounded interactions. In Theorem 3.1, we demonstrate that thefinite volume Heisenberg dynamics τΛt,s, as defined in (3.6), associated to any Φ ∈ BF (I) satisfies aLieb-Robinson bound. Such bounds provide an estimate for the speed of propagation of dynamicallyevolved observables in a quantum lattice system. One can use these bounds to show that for smalltimes the dynamically evolved observable is well approximated by a local operator. For this reason,Lieb-Robinson bounds and other similar results are often referred to as quasi-locality estimates.

Before we state the result, two more pieces of notation will be useful. First, to each X ∈ P0(Γ),we denote by ∂IΦX ⊂ X the Φ-boundary of X:

(3.20) ∂IΦX := x ∈ X : ∃Z ∈ P0(Γ) with x ∈ Z,Z ∩ (Γ \X) 6= ∅, and ∃t ∈ I with Φ(Z, t) 6= 0 .

In some estimates, it may be useful to restrict the time interval used to define the Φ-boundary.For instance, given Φ ∈ BF (R) one could find that ∂RΦX = X for some X, while ∂IΦX is strictlysmaller for a subinterval I ⊂ R. From now on we will drop the time-interval I from the notationand simply write ∂ΦX. We note also that in many situations, not much is lost by using X insteadof ∂ΦX in the following estimates.

Second, for Φ ∈ BF (I), and s, t ∈ I, the quantity It,s(Φ) defined by

(3.21) It,s(Φ) = CF

∫ max(t,s)

min(t,s)‖Φ‖F (r) dr,

will appear in many results we provide, including Theorem 3.1. Clearly, if CF ‖Φ(r)‖F ≤M , for allr ∈ [min(t, s),max(t, s)], we have It,s(Φ) ≤ |t− s|M . For example we see that

It,s(Φ) ≤ CF |t− s||||Φ|||F ,

with

(3.22) |||Φ|||F := supt∈I

‖Φ(t)‖F .

Theorem 3.1 (Lieb-Robinson Bound). Let Φ ∈ BF (I) and X,Y ∈ P0(Γ) with X ∩ Y = ∅. Forany Λ ∈ P0(Γ) with X ∪ Y ⊂ Λ and any A ∈ AX and B ∈ AY , we have

(3.23)∥

∥

[

τΛt,s(A), B]∥

∥ ≤ 2‖A‖‖B‖CF

(

e2It,s(Φ) − 1)

D(X,Y )

for all t, s ∈ I. Here, CF is the constant in (3.9), and the quantity D(X,Y ) is given by

(3.24) D(X,Y ) = min

∑

x∈X

∑

y∈∂ΦY

F (d(x, y)),∑

x∈∂ΦX

∑

y∈Y

F (d(x, y))

.

It is easy to see that with the definition F1(r) = C−1F F (r), F1 is a new F -function in terms of

which the bound (3.23) slightly simplifies in the sense that CF1 = 1. This is a general feature ofour estimates involving F -functions and the associated norms on the interactions. In the followingsections a variety of different F -functions will be used. Often, new F -functions are obtained byelementary transformations of old ones, see, e.g., Section 8.3. Instead of figuring out the normal-ization constants that make CF = 1 for each of the F -functions, we note that the final result canbe expressed with a renormalized F -function such that CF = 1.

Before moving on to the proof of the theorem, we make two simple remarks that are implicit inmany applications of the Lieb-Robinson bounds. First, one trivially has

∥

∥

[

τΛt,s(A), B]∥

∥ ≤ 2‖A‖‖B‖.Second, in the case that Φ ∈ BFg(I), for a weighted F -function Fg(r) = e−g(r)F (r), we can further


estimate

D(X,Y ) ≤ min

∑

x∈X

∑

y∈∂ΦY

F (d(x, y)),∑

x∈∂ΦX

∑

y∈Y

F (d(x, y))

e−g(d(X,Y ))

≤ min|∂ΦX|, |∂ΦY |‖F‖e−g(d(X,Y )),(3.25)

where d(X,Y ) is the distance between X and Y . When g(r) = ar for some a > 0 (i.e. Φ ∈ BFa(I)with Fa(r) = e−arF (r)) it makes sense to define the quantity va = 2a−1CFa |||Φ|||Fa

which is oftenreferred to as the Lieb-Robinson velocity, or more correctly a bound for the speed of propagation ofany type of disturbance or signal in the system. In terms of va, (3.23) implies the more transparentestimate

(3.26)∥

∥

[

τΛt,s(A), B]∥

∥ ≤ 2‖A‖‖B‖‖F‖C−1Fa

min|∂ΦX|, |∂ΦY |ea(va |t−s|−d(X,Y )).

Note that the RHS of the bounds in (3.23) and (3.26) are expressed in terms of quantities definedover the system on Γ and, in particular, these estimates are uniform in the choice of the finite setΛ ⊂ Γ. This fact will be vital in many applications.

Before we prove Theorem 3.1, we first prove a lemma. For this lemma and later use, we definethe ‘surface’ of X in the volume Λ, denoted SΛ(X), as follows:

(3.27) SΛ(X) = Z ⊂ Λ : Z ∩X 6= ∅ and Z ∩ (Λ \X) 6= ∅.It is simply the set of supports of the interaction terms that connect X and Λ \X. We will alsouse the following notation, for X,Y ∈ P0(Γ):

(3.28) δY (X) =

0 if X ∩ Y = ∅1 if X ∩ Y 6= ∅.

Lemma 3.2. Let Φ ∈ BF (I). Fix Y ∈ P0(Γ), B ∈ AY , and Λ ∈ P0(Γ) with Y ⊂ Λ. For any

X ⊂ Λ, the family mappings gX,Bt,s : AX → AΛ for t, s ∈ I, defined by

(3.29) gX,Bt,s (A) = [τΛt,s(A), B]

are norm-continuous; more precisely, (s, t) 7→ gX,Bt,s is jointly continuous in the norm on B(AX ,AΛ).

Moreover, for fixed t and s, the mapping gX,Bt,s satisfies

(3.30) ‖gX,Bt,s ‖ ≤ 2‖B‖δY (X) + 2

∑

Z∈SΛ(X)

∫ max(t,s)

min(t,s)‖gZ,Br,s (Φ(Z, r))‖ dr.

The continuity of gX,Bt,s follows directly from the joint norm continuity of (s, t) 7→ UΛ(t, s) as

proven in Proposition 2.2 (v), see also statements following. In fact, for any A ∈ AX , one has thesimple estimate

(3.31) ‖gX,Bt,s (A)− gX,B

t0,s0(A)‖ ≤ 2‖B‖‖τΛt,s(A) − τΛt0,s0(A)‖ ≤ 4‖A‖‖B‖‖UΛ(t, s)− UΛ(t0, s0)‖ .In general, this continuity does not carry over to the thermodynamic limit. Of course, we alwayshave

(3.32) ‖gX,Bt,s (A(t))‖ ≤ ‖gX,B

t,s ‖ ‖A(t)‖.

We also note that the map gX,Bt,s equals the restriction of gΛ,Bt,s to AX . It is useful, however, to

consider them as separate maps for each X ⊂ Λ, because the estimates for their norms depend

crucially on X through SΛ(X). Also note that each gX,Bt,s only depends on interaction terms Φ(Z, r)

such that Z ⊂ Λ and r ∈ [min(t, s),max(t, s)].


Proof of Lemma 3.2. Fix X ⊂ Λ, A ∈ AX , and s ∈ I. Recall that the inverse dynamics is given by

(3.33) τXt,s(A) = UX(t, s)AUX(t, s)∗,

where the unitary mappings UX(t, s) are defined as in (3.5), see also (3.4), with Λ = X. Consider

the function fs : I → AΛ given by fs(t) = gX,Bt,s (τXt,ss(A)). It follows that fs(t) = [τΛt,s τXt,s(A), B]

is strongly differentiable in t and a short calculation shows that

d

dtfs(t) = i

[

τΛt,s([

HΛ(t)−HX(t), τXt,s(A)])

, B]

= i∑

Z∈SΛ(X)

[[

τΛt,s(Φ(Z, t)), τΛt,s τXt,s(A)

]

, B]

= i∑

Z∈SΛ(X)

[

τΛt,s(Φ(Z, t)), fs(t)]

− i∑

Z∈SΛ(X)

[

τΛt,s τXt,s(A),[

τΛt,s(Φ(Z, t)), B]]

,(3.34)

where: for the first equality we have used that the adjoint of the unitary propagator has astrong derivative which can be calculated using (2.18), for the second equality we have used thatsupp(τXt,s(A)) ⊂ X, and for the last equality we used the Jacobi identity. Hence,

(3.35)d

dtfs(t) = −i[C(t), fs(t)] +D(t)

where

(3.36) C(t) = −∑

Z∈SΛ(X)

τΛt,s(Φ(Z, t)) and D(t) = −i∑

Z∈SΛ(X)

[

τΛt,s τXt,s(A),[

τΛt,s(Φ(Z, t)), B]]

.

Since C and D are finite sums and products of strongly continuous functions with C(t) = C(t)∗,they satisfy the assumptions on A and B, respectively, in Lemma 2.3 with t0 = s. Thus, we have

(3.37)∥

∥

[

τΛt,s τXt,s(A), B]∥

∥ ≤ ‖[A,B]‖ + 2‖A‖∑

Z∈SΛ(X)

∫ max(t,s)

min(t,s)

∥

∥

[

τΛr,s(Φ(Z, r)), B]∥

∥ dr.

As fs(t) = gX,Bt,s (τXt,s(A)), the bound claimed in (3.30) follows by applying (3.37) to A = τXt,s(A).

Proof of Theorem 3.1. Below, we will prove that ‖[τΛt,s(A), B]‖ satisfies the estimate (3.23) with

(3.38) D(X,Y ) =∑

x∈∂ΦX

∑

y∈Y

F (d(x, y)).

Since we also have that

(3.39) ‖[τΛt,s(A), B]‖ = ‖τΛt,s(

[A, τΛt,s(B)])

‖ = ‖[τΛs,t(B), A]‖ ,the bound in (3.23) with D(X,Y ) defined to be the minimum in (3.24) is also clear.

Let X, Y , Λ, A, and B be as in Theorem 3.1. An application of Lemma 3.2 demonstrates that

(3.40) ‖[τΛt,s(A), B]‖ ≤ 2‖A‖‖B‖δY (X) + 2‖A‖∑

Z∈SΛ(X)

∫ max(s,t)

min(s,t)‖[τΛr,s(Φ(Z, r)), B]‖ dr

for all s, t ∈ I. As such, it suffices to consider the case s ≤ t. Applying the bound (3.32) to theintegrand in (3.40), it is clear that we may iteratively apply Lemma 3.2. As a result, for any N ≥ 1

(3.41) ‖[τΛt,s(A), B]‖ ≤ 2‖A‖‖B‖(

δY (X) +N∑

n=1

an(t)

)

+RN+1(t)


where

an(t) = 2n∑

Z1∈SΛ(X)

∑

Z2∈SΛ(Z1)

· · ·∑

Zn∈SΛ(Zn−1)

δY (Zn)

∫ t

s

∫ r1

s· · ·∫ rn−1

s×

×

n∏

j=1

‖Φ(Zj , rj)‖

drndrn−1 · · · dr1(3.42)

and

RN+1(t) = 2N+1∑

Z1∈SΛ(X)

∑

Z2∈SΛ(Z1)

· · ·∑

ZN+1∈SΛ(ZN )

∫ t

s

∫ r1

s· · ·∫ rN

s×

×

N∏

j=1

‖Φ(Zj , rj)‖

‖[τΛrN+1,s(Φ(ZN+1, rN+1)), B]‖drN+1drN · · · dr1.(3.43)

The remainder term RN+1(t) is estimated as follows. First, we observe that

(3.44) ‖[τΛrN+1,s(Φ(ZN+1, rN+1)), B]‖ ≤ 2‖B‖ ‖Φ(ZN+1, rN+1)‖ .

Next, we note that the sums above are in fact sums over chains of sets (Z1, Z2, · · · , ZN+1) whichsatisfy Z1 ∩ ∂ΦX 6= ∅ and Zj ∩ Zj−1 6= ∅ for 2 ≤ j ≤ N + 1. As such, there are pointsw1, w2, · · · , wN+1 ∈ Λ such that w1 ∈ Z1 ∩ ∂ΦX and wj ∈ Zj ∩ Zj−1 for all 2 ≤ j ≤ N + 1.A simple upper bound on these sums is then obtained by overcounting:

(3.45)∑

Z1∈SΛ(X)

∑

Z2∈SΛ(Z1)

· · ·∑

ZN+1∈SΛ(ZN )

∗ ≤∑

w1∈∂ΦX

∑

w2,...,wN+2∈Λ

∑

Z1,...,ZN+1⊂Λ:

wk,wk+1∈Zk,k=1,...,N+1

∗

where ∗ denotes arbitrary non-negative quantities. We have also used that the set ZN+1 mustcontain more than one point since ZN+1 ∈ SΛ(ZN ). As Φ ∈ BF (I), (3.19) implies that

(3.46)∑

Zk⊂Λ:

wk,wk+1∈Zk

‖Φ(Zk, rk)‖ ≤ ‖Φ‖F (rk)F (d(wk, wk+1))

for each 1 ≤ k ≤ N + 1. Using this bound as well as (3.8) and (3.9), we conclude that

RN+1(t) ≤ 2‖B‖2N+1

∫ t

s· · ·∫ rN

s

∑

w1∈∂ΦX

∑

w2,...,wN+2∈Λ

∑

Z1,...,ZN+1⊂Λ:

wk,wk+1∈Zk,k=1,...,N+1

N+1∏

j=1

‖Φ(Zj , rj)‖drN+1 · · · dr1

≤ 2‖B‖2N+1

∫ t

s· · ·∫ rN

s

∑

w1∈∂ΦX

∑

w2,...,wN+2∈Λ

N+1∏

j=1

‖Φ‖F (rj)F (d(wj , wj+1))drN+1 · · · dr1

≤ 2‖B‖2N+1CNF

∑

w1∈∂ΦX

∑

wN+2∈Λ

F (d(w1, wN+2))

∫ t

s· · ·∫ rN

s

N+1∏

j=1

‖Φ‖F (rj)drN+1 · · · dr1

≤ 2‖B‖|∂ΦX|‖F‖CF

(

2CF

∫ ts ‖Φ‖F (r) dr

)N+1

(N + 1)!.(3.47)

We note that, in the last inequality, we performed the integration over the simplex. Since ‖Φ‖F islocally integrable on I, this remainder clearly goes to 0 as N → ∞.


A similar estimate can be applied to the terms an(t). In fact, these terms are also sums overchains, however, there is a restriction: only those chains whose final link Zn satisfies Zn ∩ Y 6= ∅contribute to the sum. Recalling that It,s(Φ) = CF

∫ ts ‖Φ‖F (r)dr, the bound

(3.48) an(t) ≤1

CF

(2It,s(Φ))n

n!

∑

x∈∂ΦX

∑

y∈Y

F (d(x, y))

then follows as above. Since δY (X) = 0 and n ≥ 1, the bound in (3.23) is now clear.

3.2. A class of unbounded Hamiltonians. As we now discuss, the methods in the previoussubsection extend to models with unbounded on-site terms. Consider a quantum lattice systemcomprised of (Γ, d) and AΓ. Let I ⊂ R be an interval, F an F -function on (Γ, d), and Φ ∈ BF (I)a time-dependent interaction. To each z ∈ Γ, fix a self-adjoint operator Hz with dense domainDz ⊂ Hz. For any Λ ∈ P0(Γ) and t ∈ I, consider the finite-volume Hamiltonian

(3.49) HΛ(t) =∑

z∈Λ

Hz +∑

Z⊂Λ

Φ(Z, t).

The non-interacting Hamiltonian

(3.50) H(0)Λ =

∑

z∈Λ

Hz

is essentially self-adjoint with domain

(3.51) DΛ = span⊗

z∈Λ

ψz |ψz ∈ Dz for all z ∈ Λ,

see [110, Theorem VIII.33 and Corollary]. Since the time-dependent terms are bounded, it followsfrom [126, Theorem 5.28] that for each t ∈ I, HΛ(t) is essentially self-adjoint on HΛ with domain

DΛ. We proceed by using the notation H(0)Λ and HΛ(t) for the corresponding self-adjoint closures.

As Φ ∈ BF (I), it is a strongly continuous interaction, and so for any Λ ∈ P0(Γ) Proposition 2.4guarantees the existence of a finite volume unitary propagator corresponding to HΛ(t). Let usbriefly review this in order to motivate our definition of the finite volume dynamics. By Stone’s

theorem, the non-interacting self-adjoint Hamiltonian H(0)Λ generates a free-dynamics

(3.52) τ(0)t (A) = eitH

(0)Λ Ae−itH

(0)Λ for all A ∈ AΛ and all t ∈ R

in terms of a group of strongly continuous unitaries U(0)Λ (t, 0) = e−itH

(0)Λ . In this case,

(3.53) HΛ(t) =∑

Z⊂Λ

τ(0)t (Φ(Z, t)) for all t ∈ I,

is pointwise self-adjoint with HΛ : I → AΛ strongly continuous. By Proposition 2.2 (v), there is aunique strong solution of the initial value problem

(3.54)d

dtUΛ(t, s) = −iHΛ(t)UΛ(t, s) with UΛ(s, s) = 1

for each s ∈ I. In terms of these solutions, we introduce

(3.55) UΛ(t, s) = e−itH(0)Λ UΛ(t, s)e

isH(0)Λ ,

for any s, t ∈ I. As is demonstrated in the proof of Proposition 2.4, the operators UΛ(t, s)s,t∈Iform a two-parameter family of unitaries. They satisfy the co-cycle property (2.19), and generate


the unique locally norm bounded weak solutions of the time-dependent Schrodinger equation cor-responding to HΛ(t). We use these unitaries to define a dynamics associated to HΛ(t); namely forany s, t ∈ I, we take τΛt,s : AΛ → AΛ as

(3.56) τΛt,s(A) = UΛ(t, s)∗AUΛ(t, s).

One readily checks that the family τΛt,s | t, s ∈ I of automorphisms on AΛ satisfies the co-cycleproperty and that the following analogue of Theorem 3.1 holds for this dynamics.

Theorem 3.3. Let Hzz∈Γ be a collection of densely defined self-adjoint operators, Φ ∈ BF (I),and τΛt,s be the dynamics given in (3.56). Let X,Y ∈ P0(Γ) be disjoint sets. For any Λ ∈ P0(Γ)with X ∪ Y ⊂ Λ and any A ∈ AX and B ∈ AY , the bound

(3.57)∥

∥

[

τΛt,s(A), B]∥

∥ ≤ 2‖A‖‖B‖CF

(

e2It,s(Φ) − 1)

D(X,Y )

holds for all t, s ∈ I. Here, CF is the constant in (3.9), and the quantities It,s(Φ) and D(X,Y ) areas discussed earlier; see (3.21) and (3.24), respectively.

Proof. By construction, it is clear that

(3.58) τΛt,s(A) = τ (0)s τΛt,s τ(0)t (A) .

Here τ(0)s and τΛt,s are the inverse free-dynamics and the interaction-picture dynamics, i.e.

(3.59) τ (0)s (A) = U(0)Λ (s, 0)AU

(0)Λ (s, 0)∗ and τΛt,s(A) = UΛ(t, s)

∗AUΛ(t, s) .

In this case,

(3.60) ‖[τΛt,s(A), B]‖ = ‖[τΛt,s(τ (0)t (A)), τ (0)s (B)]‖ .

Note that for all t ∈ I, τ(0)t (A) ∈ AX and τ

(0)t (B) ∈ AY . Moreover, the interaction-picture dynamics

is generated by the strongly continuous interaction Φ with terms

(3.61) Φ(Z, t) = eitH(0)Z Φ(Z, t)e−itH

(0)Z

for any Z ∈ P0(Γ) and t ∈ I. Since Φ(Z, t) and Φ(Z, t) have the same support and the same norm,

it is clear that ‖Φ‖F (t) = ‖Φ‖F (t) for all t ∈ I. In this case, the bound in (3.57) follows from

Theorem 3.1 applied to the interaction-picture dynamics Φ.

In Section 2.3.2, we considered a family of Hamiltonians, see (2.46), with bounded interactionswhich depend not only on time but also on an auxillary parameter. Within the context of quantumlattice models, the corresponding finite-volume Hamiltonian may have the form

(3.62) HλΛ(t) =

∑

z∈Λ

Hz +∑

X⊂Λ

Φλ(Z, t).

If (λ, t) 7→ Φλ(X, t) is jointly strongly continuous and strongly differentiable with respect to λ,Proposition 2.7 applies to the corresponding finite-volume dynamics

(3.63) τΛ,λt,s (A) = UλΛ(t, s)

∗AUλΛ(t, s) .

To be clear, we note that the unitaries UλΛ(t, s), in (3.63) above, are constructed as in (3.55) by

first solving the analogue of (3.54) with

(3.64) HλΛ(t) =

∑

Z⊂Λ

τ(0)t (Φλ(Z, t)) for all t ∈ I .

With no further assumptions on the interaction terms, the bound provided by Proposition 2.7 on

the λ-derivative of τΛ,λt,s (A), see (2.59), will generally depend on the volume Λ. However, under the


additional assumption that Φλ,Φ′λ ∈ BF (I), one obtains a better, volume independent estimate on

the derivative. In fact, arguing as in the proof of Proposition 2.7 one finds that if s ≤ t, then

(3.65)d

dλτΛ,λt,s (A) = i

∑

Z⊂Λ

∫ t

sτΛ,λr,s ([Φ′

λ(Z, r), τΛ,λt,r (A)]) dr;

compare with (2.60). Since Φλ ∈ BF (I), the Lieb-Robinson bound (3.57) holds for the dynamics

τΛ,λt,s . In this case, an application of Corollary 8.5 shows that for all A ∈ AX

(3.66)

∥

∥

∥

∥

d

dλτΛ,λt,s (A)

∥

∥

∥

∥

≤ 2‖A‖‖F‖|X|e2It,s(Φλ)

∫ t

s‖Φ′

λ‖F (r)dr ,

the right-hand-side of which is independent of Λ.

3.3. The infinite-volume dynamics. In this section, we will prove several convergence and con-tinuity results for the Heisenberg dynamics associated with interactions Φ ∈ BF (I) that make useof Lieb-Robinson bounds. As is well-known, see e.g. [20], Lieb-Robinson bounds can be used toprove the existence of a dynamics in the thermodynamic limit for sufficiently short-range interac-tions. In Theorem 3.5, we show that given an interaction Φ ∈ BF (I) the dynamics correspondingto finite-volume restrictions of Φ converge in the thermodynamic limit. To prove the existence ofthe thermodynamic limit, we will apply Theorem 3.4 below, which establishes that the Heisenbergdynamics is continuous in the interaction space. For example, in the case of time-independentinteractions, Theorem 3.4 implies that the difference between the dynamical evolution of a localobservable A with respect to two different interactions Φ,Ψ ∈ BF is small if ‖Φ − Ψ‖F is small.The statement of this result for finite-volume Heisenberg dynamics is the content of Theorem 3.4,with the analogous thermodynamic limit statement given in Corollary 3.6. Lastly, given a sequenceof interactions which converge locally in F -norm, see Definition 3.7 below, we show that the corre-sponding dynamics (which necessarily exist by Theorem 3.5) converge as well; this is the contentof Theorem 3.8. In particular, this can be used to prove that the thermodynamic limit of theHeisenberg dynamics is unchanged by the addition of (sufficiently local) boundary conditions. Ifthe interactions are norm continuous, the cocycle of automorphisms describing the infinite-volumedynamics is differentiable with a strongly continuous generator. This is shown in Theorem 3.9.

We now begin with the continuity statement. For this result, we will once again make use of thequantity It,s(Φ), which is defined in (3.21).

Theorem 3.4. Consider a quantum lattice system comprised of (Γ, d) and AΓ. Let I ⊂ R be aninterval, F be an F -function on (Γ, d), and Φ,Ψ ∈ BF (I) be time-dependent interactions. Fix acollection of densely defined, self-adjoint on-site Hamiltonians Hzz∈Γ, and for any Λ ∈ P0(Γ),define Hamiltonians

(3.67) H(Φ)Λ (t) =

∑

z∈Λ

Hz +∑

Z⊂Λ

Φ(Z, t) and H(Ψ)Λ (t) =

∑

z∈Λ

Hz +∑

Z⊂Λ

Ψ(Z, t)

as well as their corresponding dynamics, τΛt,s and αΛt,s, for each s, t ∈ I, respectively.

(i) For any X,Λ ∈ P0(Γ) with X ⊂ Λ, the bound

(3.68) ‖τΛt,s(A)− αΛt,s(A)‖ ≤ 2‖A‖

CFe2min(It,s(Φ),It,s(Ψ))It,s(Φ−Ψ)

∑

x∈X

∑

y∈Λ

F (d(x, y))

holds for all A ∈ AX and s, t ∈ I.(ii) For any X,Λ0,Λ ∈ P0(Γ) with X ⊂ Λ0 ⊂ Λ, the bound

(3.69) ‖τΛt,s(A)− τΛ0t,s (A)‖ ≤ 2‖A‖

CFe2It,s(Φ)It,s(Φ)

∑

x∈X

∑

y∈Λ\Λ0

F (d(x, y))

holds for all A ∈ AX and s, t ∈ I.


Using (3.8), the estimates in (3.68) and (3.69) can be interpreted as bounds on the norm of thedifference of two dynamics, thought of as maps from AX to AΛ, that are uniform in Λ but growlinearly in |X|. Since the dynamics are, of course, automorphisms, the bounds of Theorem 3.4 areonly nontrivial if the RHS of (3.68) and (3.69) are smaller than 2‖A‖. As is well known, this will betrue for both cases if |t− s| is sufficiently small. Additionally, the bound in (3.68) will be nontrivialif ‖Φ−Ψ‖F is small, and the bound in (3.69) will be nontrivial if d(X,Λ \Λ0) is small. Note that,even if a map is bounded on all of Aloc

Γ , as is the case in Theorem 3.4, norm bounds for their localrestriction can be very useful.

Proof. To prove (i), we first consider the case that Hz = 0 for all z ∈ Γ. For any Λ ∈ P0(Γ), X ⊂ Λ,s, t ∈ I, and A ∈ AX we write the corresponding difference as

(3.70) τΛt,s(A)− αΛt,s(A) =

∫ t

s

d

dr

(

τΛr,s αΛt,r(A)

)

dr = i

∫ t

sτΛr,s

([

H(Θ)Λ (r), αΛ

t,r(A)])

dr

where we have introduced the local Hamiltonian

(3.71) H(Θ)Λ (r) =

∑

Z⊂Λ

Θ(Z, r) with Θ(Z, r) = Φ(Z, r)−Ψ(Z, r).

Note that the equality in (3.70) is to be understood in the strong sense. When s ≤ t, a simplenorm bound shows then that

(3.72) ‖τΛt,s(A)− αΛt,s(A)‖ ≤

∑

Z⊂Λ

∫ t

s‖[αΛ

t,r(A),Θ(Z, r)]‖ dr.

An application of Corollary 8.5 with R = 0, then gives

(3.73)∑

Z⊂Λ:

d(Z,X)=0

∫ t

s‖[αΛ

t,r(A),Θ(Z, r)]‖ dr ≤ 2‖A‖CF

It,s(Θ)∑

x∈X

∑

y∈Λ

F (d(x, y))

and moreover,

(3.74)∑

Z⊂Λ:

d(Z,X)>0

∫ t

s‖[αΛ

t,r(A),Θ(Z, r)]‖ dr ≤ 2‖A‖CF

(

e2It,s(Ψ) − 1)

It,s(Θ)∑

x∈X

∑

y∈Λ\X

F (d(x, y)).

The bound (3.68) follows from observing that one could instead have estimated the differenceαΛt,s(A)− τΛt,s(A), which corresponds to exchanging τΛt,s and αΛ

t,s in (3.70)-(3.74).To extend the result to the situation with non-trivial Hz, we argue with the interaction picture

dynamics as was done in the proof of Theorem 3.3. Using (3.58), it is clear that

(3.75) ‖τΛt,s(A)− αΛt,s(A)‖ = ‖τΛt,s(τ (0)t (A)) − αΛ

t,s(τ(0)t (A))‖

and since τ(0)t (A) ∈ AX , the proof in the general situation reduces to the previous case.

The proof of (ii) is nearly identical. In fact, again in the case that Hz = 0 for all z ∈ Γ, theanalogue of (3.72) is

(3.76) ‖τΛt,s(A)− τΛ0t,s (A)‖ ≤

∑

Z∈SΛ(Λ0)

∫ t

s‖[τΛ0

t,r (A),Φ(Z, r)]‖ dr,

where we have taken advantage of cancellations and used (3.27). The bound in (3.69) now followsfrom similar estimates to those used in Proposition 8.4 and Corollary 8.5.

A first application of Theorem 3.4 is a proof that given any collection of self-adjoint on-sitesHzz∈Γ and an interaction Φ ∈ BF (I) there is a corresponding infinite-volume dynamics on AΓ.We obtain this infinite-volume dynamics as a limit of finite-volume dynamics. With this in mind,


we will say that a sequence Λnn≥1 ⊂ P0(Γ) is increasing and exhaustive if Λn ⊂ Λn+1 for alln ≥ 1 and given any X ∈ P0(Γ), there is an N ≥ 1 for which X ⊂ ΛN .

Theorem 3.5. Under the assumptions of Theorem 3.4, for each A ∈ AlocΓ and any s, t ∈ I,

(3.77) τt,s(A) = limΛ↑Γ

τΛt,s(A)

exists in norm and the convergence is uniform for s and t in compact subsets of I. The limit maybe taken along any increasing, exhaustive sequence of finite subsets of Γ, and is independent of thesequence. Moreover, τt,s is a co-cycle of automorphisms of AΓ. If there exists M ≥ 0 such that‖Hz‖ ≤M for all z ∈ Γ, then (t, s) 7→ τt,s(A) is norm continuous for all A ∈ AΓ.

For unbounded Hz, the continuity of τt,s is limited by the continuity of the on-site dynamics

τ(0)t . In a suitable representation of AΓ on a Hilbert space, one can retrieve weak continuity of thedynamics. See [90] for an example. Continuity properties of τt,s and other families of maps will befurther discussed in Section 4.2.1.

Proof. Let X ∈ P0(Γ), A ∈ AX , and consider Λnn≥0 any increasing, exhaustive sequence of finitesubsets of Γ. Since the sequence is exhaustive, there exists N ≥ 1 for which X ⊂ Λn for all n ≥ N .For any integers n,m with N ≤ m ≤ n, Theorem 3.4 (ii) implies

(3.78) ‖τΛnt,s (A)− τΛm

t,s (A)‖ ≤ 2‖A‖CF

e2It,s(Φ)It,s(Φ)∑

x∈X

∑

y∈Λn\Λm

F (d(x, y)),

where, again, It,s(Φ) is as defined in (3.21). By (3.8), the RHS converges to 0 as n, m → ∞. As

such, for any [a, b] ⊂ I, the sequence of observables τΛnt,s (A)n≥0 is Cauchy in norm, uniformly for

s, t ∈ [a, b].The proof of the remaining facts in the statement of this theorem is standard and proceeds in the

same way as is done, e.g., in [120] for quantum spin models with time-independent interactions.

Combining Theorem 3.4 and Theorem 3.5, we obtain the following useful estimates for the infinitevolume dynamics.

Corollary 3.6. Under the assumptions of Theorem 3.4,

(i) For any X, Y ∈ P0(Γ) such that X ∩ Y = ∅, the bound

‖[τt,s(A), B]‖ ≤ 2‖A‖‖B‖CF

(e2It,s(Φ) − 1)D(X,Y )

holds for all A ∈ AX , B ∈ AY , and t, s ∈ I.(ii) For any X ∈ P0(Γ), the bound

(3.79) ‖τt,s(A)− αt,s(A)‖ ≤ 2‖A‖CF

e2min(It,s(Φ),It,s(Ψ))It,s(Φ−Ψ)∑

x∈X

∑

y∈Γ

F (d(x, y))

holds for all A ∈ AX and s, t ∈ I.(iii) For any X,Λ ∈ P0(Γ) with X ⊂ Λ, the bound

(3.80) ‖τt,s(A)− τΛt,s(A)‖ ≤ 2‖A‖CF

e2It,s(Φ)It,s(Φ)∑

x∈X

∑

y∈Γ\Λ

F (d(x, y))

holds for all A ∈ AX and s, t ∈ I.

We now prove a convergence result for the dynamics associated to interactions in BF (I). First, weintroduce some notation and terminology associated with extensions and restrictions of interactions,and then state the result.


In certain applications, e.g. when considering boundary conditions, rather than considering asingle interaction Φ : P0(Γ) → Aloc

Γ , we may want to consider a family of strongly continuous

interactions ΦΛ : P0(Λ) × I → AlocΛ |Λ ∈ P0(Γ). In this situation, a single mapping ΦΛ can

be extended to an interaction on all of Γ by declaring that ΦΛ(Z, t) = 0 for any Z ∈ P0(Γ) withZ ∩ (Γ \ Λ) 6= ∅. We call this new mapping the extension of ΦΛ to Γ, and continue to denote it byΦΛ. Similarly, given ΦΛ and Λ0 ⊂ Λ, we define ΦΛ Λ0 : P0(Λ0)× I → Aloc

Λ0by

(3.81) ΦΛ Λ0 (X, t) = ΦΛ(X, t) for all X ∈ P0(Λ0) and t ∈ I

We call the mapping ΦΛ Λ0 the restriction of ΦΛ to Λ0. If the dynamics associated to ΦΛ Λ0

exists, we will denote it by τΦΛ,Λ0t,s . Of course, if Λ0 is finite, such a dynamics always exists and it

is generated by the time-dependent Hamiltonian

(3.82) HΦΛΛ0

(t) =∑

X⊂Λ0

ΦΛ(X, t) .

We now introduce the notion of local convergence in F -norm.

Definition 3.7. Let (Γ, d) and AΓ be a quantum lattice system, and I ⊂ R be an interval. Wesay that a sequence of interactions Φnn≥1 converges locally in F -norm to Φ if there exists anF -function, F , such that:

(i) Φn ∈ BF (I) for all n ≥ 1,(ii) Φ ∈ BF (I),(iii) For any Λ ∈ P0(Γ) and each [a, b] ⊂ I,

(3.83) limn→∞

∫ b

a‖(Φn − Φ) Λ ‖F (t) dt = 0 .

Moreover, if F is an F -function for which (i)-(iii) are satisfied, we say that Φn converges locally inF -norm to Φ with respect to F .

Recall that a strongly continuous interaction Φ ∈ BF (I) with ‖Φ‖F : I → [0,∞) given by

(3.84) ‖Φ‖F (t) = supx,y∈Γ

1

F (d(x, y))

∑

X∈P0(Γ):

x,y∈X

‖Φ(X, t)‖

is locally bounded. In this situation, Theorem 3.5 demonstrates that there exists a co-cycle ofautomorphisms of AΓ, which we denote by τΦt,s and refer to as the dynamics associated to Φ. Notethat we have taken the self-adjoint on-sites terms to be identically zero, i.e. Hz = 0 for all z ∈ Γ;see comments following the proof of Theorem 3.8.

Theorem 3.8. Let Φnn≥1 be a sequence of time-dependent interactions on Γ with Φn converginglocally in the F -norm to Φ with respect to F .

(i) If for every [a, b] ⊂ I,

(3.85) supn≥1

∫ b

a‖Φn‖F (t) dt <∞ ,

then, for any X ∈ P0(Γ), A ∈ AX , and s, t ∈ I, s ≤ t, we have convergence of the dynamics:

(3.86) limn→∞

‖τΦnt,s (A)− τΦt,s(A)‖ = 0.

Moreover, the convergence is uniform for s, t in compact intervals, and the dynamics iscontinuous:

(3.87) ‖τΦt,s(A)−A‖ ≤ 2|X|‖A‖‖F‖∫ t

s‖Φ‖F (r)dr.


(ii) If in addition, for all Λ ∈ P0(Γ), and r ∈ I, we also have pointwise local convergence:

(3.88) limn→∞

‖(Φn − Φ) Λ ‖F (r) = 0,

and uniform boundedness of the interactions on compact intervals I0 ⊂ I:

(3.89) supn

supt∈I0

‖Φn‖F (t) <∞,

then, the generators converge uniformly for t in compacts: for all compact I0 ⊂ I andA ∈ Aloc

Γ , we have

(3.90) limn

supt∈I0

‖δ(n)t (A)− δt(A)‖ = 0,

whereδ(n)r (A) =

∑

Z∈P0(Γ)

[Φn(Z, r), A], δr(A) =∑

Z∈P0(Γ)

[Φ(Z, r), A].

Proof. (i). Let X ∈ P0(Γ) and [a, b] ⊂ I be fixed. For any Λ ∈ P0(Γ) with X ⊂ Λ, define therestricted interactions Φn Λ and Φ Λ. By Theorem 3.5, a dynamics may be associated to each ofthese interactions and the estimate

‖τΦnt,s (A) − τΦt,s(A)‖ ≤ ‖τΦn

t,s (A)− τΦn,Λt,s (A)‖

+‖τΦn,Λt,s (A)− τΦ,Λ

t,s (A)‖ + ‖τΦ,Λt,s (A)− τΦt,s(A)‖(3.91)

holds for all A ∈ AX and s, t ∈ [a, b].An application of Corollary 3.6 (iii), shows that

(3.92) ‖τΦnt,s (A)− τΦn,Λ

t,s (A)‖ ≤ 2‖A‖CF

e2It,s(Φn)It,s(Φn)∑

x∈X

∑

y∈Γ\Λ

F (d(x, y))

and similarly

(3.93) ‖τΦt,s(A)− τΦ,Λt,s (A)‖ ≤ 2‖A‖


∑

x∈X

∑

y∈Γ\Λ

F (d(x, y)).

For the middle term above, we apply Theorem 3.4 (i) to find that

(3.94) ‖τΦn,Λt,s (A)− τΦ,Λ

t,s (A)‖ ≤ 2‖A‖|X|‖F‖CF

e2min(It,s(Φn),It,s(Φ))It,s((Φn − Φ) Λ).

By assumption (3.85), it is clear that supn It,s(Φn) is finite for all s, t ∈ [a, b]. In this case, forany ǫ > 0, the estimates in (3.92) and (3.93) can be made arbitrarily small for all n sufficientlylarge (for example, less than ǫ/3) with a sufficiently large, but finite choice of Λ ⊂ Γ. For any suchchoice of Λ, the bound (3.94) can be made equally small with large n by using local convergencein F -norm.

To prove the bound (3.87), we use the convergence established above, the existence of the ther-modynamic limit (Theorem 3.5), and the differentiability of the finite-volume dynamics, as follows:

‖τΦt,s(A)−A‖ = limΛ

limn

‖τΦn,Λt,s (A)−A‖

≤ lim supΛ

limn

∑

Z⊂Λ

‖∫ t

sτΦn,Λt,s ([Φn(Z, r), A])dr‖

≤ lim supΛ

lim supn

2|X|‖A‖‖F‖∫ t

s‖Φn Λ ‖F (r)dr

≤ 2|X|‖A‖‖F‖[

lim supΛ

∫ t

s‖Φ Λ ‖F (r)dr + lim sup

Λlim sup

n

∫ t

s‖(Φ− Φn) Λ ‖F (r)dr

]

.


The second term between the square brackets vanishes because the interactions converge locally inF -norm and the first term gives the desired estimate.

(ii) The interactions Φn(t) and Φ(t) have finite F -norm. Hence the corresponding derivations,

δ(n)t and δt, are well-defined on Aloc

Γ . For any Λ ∈ P0(Γ) we then have

(3.95) δ(n)t (A)− δt(A) =

∑

Z⊂Λ

[Φn(Z, t) − Φ(Z, t), A] +∑

Z,Z∩Γ\Λ 6=∅

[Φn(Z, t) − Φ(Z, t), A].

Therefore, applying (8.41) with R = 0 to the first term and using a similar argument for the secondterm, we get(3.96)

‖δ(n)t (A)− δt(A)‖ ≤ 2|X|‖A‖‖F‖‖(Φn − Φ) Λ ‖F (t) + 2|X|‖A‖‖Φn − Φ‖F (t)∑

x∈X,y∈Γ\Λ

F (x, y).

The first term on the RHS vanishes in the limit n→ ∞. Therefore

(3.97) lim supn

‖δ(n)t (A)− δt(A) ≤ 2‖|X|‖A‖(supn

‖Φn −Φ‖F (t))∑

x∈X,y∈Γ\Λ

F (x, y).

By taking Λ ↑ Γ, the convergence of the generators now follows. The estimate (3.96) shows thatthe convergence is uniform for t in a compact interval.

The dynamics considered in the proof of Theorem 3.8 above corresponds to the one whoseexistence is established in the proof of Theorem 3.5 in the special case that the on-sites Hz = 0 forall z ∈ Γ. By going to the interaction picture, as is done in the proof of Theorem 3.4, it is clear thatthe convergence results (3.87) and (3.90) hold in the case of arbitrary self-adjoint on-site terms.

Theorem 3.8 establishes sufficient conditions for the convergence of the sequence of co-cycles τΦnt,s

to the co-cycle τΦt,s, as well as the convergence of the generators δ(n)t to densely defined derivations

δt. These conditions are by no means necessary, but will serve our purposes well.We may now ask whether the dynamics satisfies additional properties and, in particular, whether

it is differentiable with derivative given by the derivation δt. The following theorem addresses thisquestion.

Theorem 3.9. For all t in an interval I, let Φ(t) ∈ BF be interactions such that t 7→ Φ(Z, t) isnorm-continuous for all Z ∈ P0(Γ), and such that ‖Φ(t)‖F is bounded on all compact intervalsI0 ⊂ I. Let τt,s denote the strongly continuous dynamics generated by Φ(t). Define, for all t ∈ I,

(3.98) δt(A) =∑

Z∈P0(Γ)

[Φ(Z, t), A], A ∈ AlocΓ .

Then, t→ δt(A) is norm-continuous for all A ∈ AlocΓ , and for all s, t ∈ I, s < t,

(3.99)d

dtτt,s(A) = iτt,s(δt(A)), A ∈ Aloc

Γ .

Proof. First, note that the conditions of Theorem 3.8, parts (i) and (ii), are satisfied for the sequenceΦn = Φ Λn , associated to any sequence of increasing and absorbing finite volumes Λn. Therefore,we have

(3.100) ‖τt,s(A)−A‖ ≤ 2|X|‖A‖‖F‖|t − s| supr∈I0

‖Φ(r)‖F , for all s, t ∈ I0

and

(3.101) δt(A) = limn

∑

Z⊂Λn

[Φ(Z, t), A], A ∈ AlocΓ .


To prove the continuity of δt(A) as a function of t, note that for s, t ∈ I,X ∈ P0(Γ), A ∈ AX , andΛ ⊂ Γ, we have

‖δt(A)− δs(A)‖ =

∥

∥

∥

∥

∥

∥

∥

∥

∑

Z∈P0(Γ)Z∩X 6=∅

[Φ(Z, t) −Φ(Z, s), A]

∥

∥

∥

∥

∥

∥

∥

∥

≤ 2‖A‖

∑

Z⊂ΛZ∩X 6=∅

‖Φ(Z, t) − Φ(Z, s)‖ +∑

Z∈P0(Γ)Z∩(Γ\Λ)6=∅,Z∩X 6=∅

‖Φ(Z, t)‖ + ‖Φ(Z, s)‖

≤ 2‖A‖

∑

Z⊂ΛZ∩X 6=∅

‖Φ(Z, t) − Φ(Z, s)‖ + 2 supr∈I0

‖Φ(r)‖F∑

x∈X

∑

y∈Γ\Λ

F (d(x, y))

.

Continuity follows from this estimate by first choosing Λ large enough to make the second termsmall and then, for that Λ, using the fact that the first term is a finite sum of continuous functionsthat vanish for t = s.

To prove differentiability, we first show that the finite-volume derivatives converge to the desiredlimit, uniformly on compact intervals. Consider

(3.102) τt,s(δt(A))− τ(n)t,s (δ

(n)t (A)) = τt,s(δt(A)− δ

(n)t (A)) + (τt,s − τ

(n)t,s )(δ

(n)t (A)).

For the first term on the RHS we use the uniformity of the convergence of the derivation ob-tained in Theorem 3.8 (ii), and the boundedness of τt,s, and we estimate the second term usingCorollary 3.6 (iii). This produces

(3.103) ‖(τt,s − τ(n)t,s )(δ

(n)t (A))‖ ≤ 2‖δ(n)t (A)‖


∑

x∈X

∑

y∈Γ\Λn

F (d(x, y)).

Since ‖δ(n)t (A)‖ is uniformly bounded in t and n, the RHS vanishes as n → ∞, uniformly fors, t in any compact interval I0. This shows the uniform convergence of the derivative, i.e. for allX ∈ P0(Γ) and A ∈ AX

(3.104) limn

sups,t∈I0

‖τt,s(δt(A)) − τ(n)t,s (δ

(n)t (A))‖ = 0.

By the continuity of τt,s(δt(A)) in t and the usual argument using the fundamental theorem ofcalculus, it now follows that τt,s(A) is differentiable in t with derivative given by iτt,s(δt(A)).

We conclude this section with a few comments.It is clear how to modify Definition 3.7 in such a way to describe sequences that are locally Cauchy

in F -norm. Given any such Cauchy sequence which also satisfies (3.85), an ǫ/3-argument almostidentical to the one in the proof of Theorem 3.8 shows that the corresponding dynamics convergeto a co-cycle of automorphisms of AΓ. With this understanding, one sees that Theorem 3.8 impliesTheorem 3.5. In fact, let Φ ∈ BF (I) and take Λnn≥1 to be an increasing, exhaustive sequenceof finite subsets of Γ. Define the sequence of interactions Φn = Φ Λn and extend Φn to all of Γas indicated above. In this case, it is clear that Φnn≥1 is locally Cauchy in the F -norm definedby F and moreover, (3.85) holds. Thus the corresponding dynamics converge. Since the sequenceΦnn≥1 also converges locally to Φ in the F -norm defined by F , we know what the generator ofthe limiting dynamics is by Theorem 3.8. In this manner we recover the fact that the limiting


dynamics is independent of the increasing, exhaustive sequence of finite-volumes. Moreover, wealso see that this limiting dynamics is invariant under a class of finite-volume boundary conditions.

As a final comment we note that one can easily find conditions under which the Duhamel formula(3.66) is also inherited by the infinite-volume dynamics. As this will not be needed in this work,we do not discuss this further.

4. Local approximations

In the previous section, we proved a Lieb-Robinson bound for the finite volume dynamics gener-ated by an interaction Φ ∈ BF (I). Such bounds provide an estimate for the speed of propagationin a quantum lattice system. More specifically, such bounds can be used to show that while thesupport of a local observable evolved under the Heisenberg dynamics is non-local, at any fixed timet, the observable essentially acts as the identity outside of a finite region of space. It is often usefulto approximate these dynamically evolved observables by strictly local observables. It is furtherdesirable that the operation of taking these local approximations has good continuity properties.This is the topic of this section.

4.1. Local approximations of observables. We first review how the support of a local observ-able can be identified using commutators. For any Hilbert space H, the algebra B(H) has a trivialcenter; in the case of a finite-dimensional Hilbert space this is known as Schur’s Lemma. A firstgeneralization of this fact is that for any two Hilbert spaces H1 and H2, the commutant of B(H1)⊗1in B(H1 ⊗H2) is given by 1⊗B(H2) (see, e.g., [63, Chapter 11]). Given this, and the structure ofthe quantum models introduced in Section 3.1.1, one concludes the following: given Λ ∈ P0(Γ) andX ⊂ Λ, if A ∈ AΛ satisfies [A,B] = 0 for all B ∈ AΛ\X , then A ∈ AX . In other words, vanishingcommutators can be used to identify the support of local observables. If the commutator [A,B] issmall but not necessarily vanishing (in norm), the following lemma, which is proved in [91], showsthat A can be well-approximated (up to error ǫ) by an observable A′ ∈ AX .

Lemma 4.1. Let H1 and H2 be complex Hilbert spaces. There is a completely positive linear mapE : B(H1 ⊗H2) → B(H1) with the following properties:

(i) For all A ∈ B(H1), E(A⊗ 1) = A;(ii) Whenever A ∈ B(H1 ⊗H2) satisfies the commutator bound

‖[A,1 ⊗B]‖ ≤ ǫ‖B‖ for all B ∈ B(H2),

E(A) satisfies the estimate

(4.1) ‖E(A)⊗ 1−A‖ ≤ ǫ;

(iii) For all C,D ∈ B(H1) and A ∈ B(H1 ⊗H2), we have

E((C ⊗ 1)A(D ⊗ 1)) = CE(A)D.

A completely positive linear map E with the properties (i) and (iii) is called a conditional ex-pectation (see, e.g., [103, Section 9.2]). If H2 is finite-dimensional, one can take E = id⊗ tr, wheretr denotes the normalized trace over H2. In this case, it is straightforward to verify the propertieslisted in the lemma (see, e.g., [23,88]). For general H2, a normalized trace does not exist, but usingLemma 4.1 it is easy to show that, at the cost of a factor 2 in the RHS of (4.1), we can replace E

with id⊗ ρ for an arbitrary state ρ on B(H2). This is the content of the following lemma from [91].

Lemma 4.2. Let H1 and H2 be two complex Hilbert spaces and ρ a state on B(H2). The mapΠρ = id ⊗ ρ satisfies properties (i) and (iii) of Lemma 4.1. Moreover, if A ∈ B(H1 ⊗H2) is suchthat there is an ǫ ≥ 0 for which

(4.2) ‖[A,1 ⊗B]‖ ≤ ǫ‖B‖ for all B ∈ B(H2),


then

(4.3) ‖Πρ(A)⊗ 1−A‖ ≤ 2ǫ .

Proof. It is clear that Πρ satisfies properties (i) and (iii) in Lemma 4.1. Moreover, since ‖Πρ‖ = 1,we also have that for any A ∈ B(H1 ⊗H2)

‖Πρ(A)⊗ 1− E(A)⊗ 1‖ = ‖Πρ(A)− E(A)‖= ‖Πρ(A− E(A)⊗ 1)‖≤ ǫ(4.4)

where we have used (4.1). The estimate (4.3) now follows:

(4.5) ‖Πρ(A)⊗ 1−A‖ ≤ ‖Πρ(A)⊗ 1− E(A)⊗ 1‖+ ‖E(A)⊗ 1−A‖ ≤ 2ǫ .

Note that the state ρ in Lemma 4.2 is not required to be normal, i.e. defined by a density matrix.However, in applications it will often be useful to take ρ normal (or locally normal in the case ofinfinite systems, see Section 4.2). For normal ρ, we can give an explicit expression for Πρ(A) in termsof its matrix elements. Since ρ is given by a density matrix, there is a countable set of orthonormalvectors ξn ∈ H2 and positive numbers ρn with

∑

n ρn = 1, so that ρ(A′) =∑

n≥1 ρn〈ξn, A′ξn〉 for

all A′ ∈ B(H2). The matrix elements of Πρ(A) are then given by:

(4.6) 〈η,Πρ(A)φ〉 =∑

n≥1

ρn〈η ⊗ ξn, A(φ ⊗ ξn)〉, for all A ∈ B(H1 ⊗H2), η, φ ∈ H1.

A number of further comments are in order. First, although the mapping Πρ depends on thestate ρ, the ‘error’ estimate in (4.3) is independent of ρ. Next, if H2 is finite dimensional, then ρcan be taken to be the normalized trace and we already know that the factor of two in (4.3) is notneeded. The bound for Πρ therefore appears to be non-optimal. The map E from Lemma 4.1 is onlyknown to be bounded (specifically, ‖E‖ = 1) and hence continuous with respect to the operatornorm topology. As such, it is not guaranteed that E is continuous when both the domain andco-domain are endowed with the strong operator topology. However, by choosing a normal stateρ, we get a map Πρ that is continuous on bounded subsets of the domain when both B(H1 ⊗H2)and B(H1) are endowed with their strong (or weak) operator topologies. The case of the strongoperator topology is the content of the next proposition. The case of the weak topology follows bya similar argument.

Recall that K : B(H1) → B(H2) is continuous on bounded subsets with both its domain andco-domain considered with the strong operator topology if given any bounded net Aα ∈ B(H1) thatconverges strongly to A ∈ B(H1), the net K(Aα) ∈ B(H2) converges strongly to K(A) ∈ B(H2).

Proposition 4.3. Let H1 and H2 be two complex Hilbert spaces, and ρ any normal state on B(H2).The following maps, when restricted to arbitrary bounded subsets of their domain, are continuouswhen both the domain and codomain are equipped with the strong operator topology:

(i) Πρ = id⊗ ρ : B(H1 ⊗H2) → B(H1)

(ii) Πρ : B(H1 ⊗H2) → B(H1 ⊗H2), A 7→ Πρ(A)⊗ 1.

Proof. (i) To prove that Πρ is continuous on bounded sets, WLOG, let Aα | α ∈ I be a net inthe unit ball of B(H1⊗H2) that converges to A ∈ B(H1 ⊗H2) in the strong operator topology, i.e.for all ψ ∈ H1 ⊗H2 the net Aαψ converges to Aψ with respect to the Hilbert space norm. Since‖Aα‖ ≤ 1 for all α ∈ I, we necessarily have that ‖A‖ ≤ 1. Let ξn, n ≥ 1 denote the orthonormalset of eigenvectors of ρ corresponding to its non-zero eigenvalues ρn. Let φ ∈ H1 with ‖φ‖ = 1. We


use (4.6) to show that Πρ(Aα)φ→ Πρ(A)φ. Note that

‖Πρ(A−Aα)φ‖ = supη∈H1,‖η‖=1

|〈η,Πρ(A−Aα)φ〉|

= supη∈H1,‖η‖=1

∑

n≥1

ρn|〈η ⊗ ξn, (A−Aα)(φ⊗ ξn)〉|.

For any ǫ > 0, choose N so that∑

n>N ρn < ǫ/4, and pick α0 ∈ I so that for all α ≥ α0 and anyn = 1, . . . , N ,

‖(A −Aα)(φ⊗ ξn)‖ ≤ ǫ/(2N).

Then for any α ≥ α0,

(4.7) ‖Πρ(A−Aα)φ‖ ≤∑

n≤N

‖(A−Aα)(φ⊗ ξn)‖+∑

n>N

2ρn < ǫ,

which establishes the desired continuity of Πρ.(ii) Let Aα |α ∈ I be a net in the unit ball of B(H1⊗H2) that converges to A ∈ B(H1⊗H2). By

(i), we know that Bα = Πρ(Aα) converges to B = Πρ(A) in the strong operator topology on B(H1).By Proposition 2.1(ii), see also (2.4) and the preceding discussion, it follows that Bα ⊗ 1l | α ∈ Istrongly converges to B ⊗ 1l in B(H1 ⊗H2).

4.2. Application to quantum lattice models. We now extend the results of the previous sub-section to infinite quantum lattice systems on Γ. In this setting states cannot be defined in termsof a single density matrix. Moreover, as explained below, we will want to define a consistent fam-ily of conditional expectations with values in AΛ, for all Λ ∈ P0(Γ). To this end, we consider alocally normal product state ρ: i.e. for each site x ∈ Γ we fix a normal state ρx on Ax = B(Hx),and take the unique state ρ on AΓ such that ρAΛ

=⊗

x∈Λ ρx for all finite Λ ⊂ Γ. Then, given

X ⊂ Λ ∈ P0(Γ), we define conditional expectations ΠX,Λρ : AΛ → AX similar to those in Lemma 4.2

by

(4.8) ΠX,Λρ (A) = (idX ⊗ ρΛ\X)(A) for all A ∈ AΛ .

Here, as before, we have taken idX as the identity map on AX . In our applications the dependenceof these maps on ρ is of minor consequence. Moreover, it will be convenient to view these maps aselements of B(AΛ). For these reasons, we suppress the dependence on ρ and define ΠΛ

X : AΛ → AΛ

by

(4.9) ΠΛX(A) = ΠX,Λ

ρ (A)⊗ 1Λ\X .

For fixed X, these projections are compatible in the sense that if X ∪Λ′ ⊂ Λ and Λ ∈ P0(Γ), then

(4.10) ΠΛX(A⊗ 1) = ΠΛ′

X∩Λ′(A)⊗ 1 for all A ∈ AΛ′ .

We summarize this and other consistency relations in Proposition 4.5 below. First, however, wedescribe how, given a fixed finite volume X, one can extend the maps ΠΛ

X , Λ ∈ P0(Γ), to an

operator ΠX on AlocΓ (and consequently , AΓ).

For any Λ′ ⊆ Λ, recall that we can identify AΛ′ as a sub-algebra of AΛ, and so we can write(4.10) as ΠΛ

X AΛ′= ΠΛ′

X∩Λ′ . In particular, if X ⊆ Λ′ ⊆ Λ, one has that ΠΛX AΛ′= ΠΛ′

X , from which

we see that the following map ΠX : AlocΓ → Aloc

Γ is well-defined:

(4.11) ΠX(A) := Πsupp(A)∪XX (A).

Since this map is bounded, in fact of norm one, ΠX has a unique extension to AΓ which we alsodenote by ΠX . We note that ΠX(A) = A if A ∈ AX . We refer to the family of conditionalexpectations ΠΛ

X , respectively ΠX , X ∈ P0(Γ), as a localizing family. By construction, the finite

volume local approximations ΠΛX all satisfy the conditions of Lemma 4.2. The following corollary

shows that the results of Lemma 4.2 also extend to ΠX : AΓ → AX .


Corollary 4.4. Let X ∈ P0(Γ) and ΠX : AΓ → AX be the extension of the map defined in (4.11).Suppose ǫ ≥ 0 and A ∈ AΓ are such that

(4.12) ‖[A,B]‖ ≤ ǫ‖B‖, for all B ∈ AlocΓ\X .

Then‖ΠX(A)−A‖ ≤ 2ǫ.

Proof. Let A ∈ AΓ. Then for any δ > 0 there exists Λ ∈ P0(Γ) and A′ ∈ AΛ such that ‖A−A′‖ < δ.

From (4.12) it follows that

‖[A′, B]‖ ≤ (ǫ+ 2δ)‖B‖, for all B ∈ AlocΛ\X .

Since ΠX(A′) = ΠΛX(A′), Lemma 4.2 implies

‖ΠX(A′)−A′‖ ≤ 2(ǫ+ 2δ).

Therefore,

‖ΠX (A)−A‖ ≤ ‖ΠX(A′)−A′‖+ ‖(ΠX − id)(A−A′)‖ ≤ 2(ǫ+ 2δ) + 2δ,

and since δ > 0 is arbitrary, the result follows.

We now state several consistency properties of the finite and infinite volume conditional expec-tations. To facilitate the statement of the properties, we use ΠΓ

X to denote ΠX : AΓ → AX .

Proposition 4.5. Fix a locally normal product state ρ on AΓ, and let X, Y, Λ′ ∈ P0(Γ) andΛ ∈ P0(Γ) ∪ Γ be such that X, Y and Λ′ are all subsets of Λ. The following properties hold forthe localizing maps defined with respect to ρ:

(i) If A ∈ AX , then ΠΛX(A) = A.

(ii) ΠΛX AΛ′= ΠΛ′

X∩Λ′ .

(iii) ΠΛX ΠΛ

Y = ΠΛY ΠΛ

X = ΠΛX∩Y .

(iv) If X ⊆ Λ′, then ΠΛ′

X ΠΛΛ′ = ΠΛ

X .

(v) ΠΛX(A)∗ = ΠΛ

X(A∗) for all A ∈ AΛ.

The proofs of these properties for Λ ∈ P0(Γ) are all elementary and follow from the definitionof Πρ

X,Λ, and the fact that ρ Λ is a product state. The statements for Λ = Γ follow from taking

finite volume limits Λ′ ↑ Γ of ΠΛ′

X (A) for A ∈ AlocΓ , and using the norm bound ‖ΠX‖ ≤ 1 to extend

to A ∈ AΓ in the usual manner.An immediate consequence of Proposition 4.5(i) is that

(4.13) limΛn↑Γ

ΠΛn(A) = A

for any sequence of increasing and absorbing finite volumes Λn, and any A ∈ AlocΓ . Since Aloc

Γ isdense in AΓ, (4.13) extends to any A ∈ AΓ.

With the aid of the maps ΠΛX , we construct local decompositions of any observable A ∈ AΛ. Let

X ⊂ Γ be finite. For any n ≥ 0, denote by X(n) ⊂ Γ the set

(4.14) X(n) = y ∈ Γ : there exists x ∈ X with d(x, y) ≤ n.Note that X(0) = X. For finite Λ ⊂ Γ with X ⊂ Λ and each integer n ≥ 0, we define ∆Λ

X(n) :

AΛ → AΛ by

(4.15) ∆ΛX(0) = ΠΛ

X and ∆ΛX(n) = ΠΛ

X(n)∩Λ −ΠΛX(n−1)∩Λ

for any n ≥ 1. Note that, in contrast to the maps ΠΛX , ∆Λ

X(n) does not only depend on the set X(n),

but on X and n separately. This slight abuse of notation will not lead to confusion. As discussedabove, ∆Λ

X(n) has range contained in AX(n)∩Λ. Moreover, as a difference of two projections, they

satisfy ‖∆ΛX(n)‖ ≤ 2.


Of course, as discussed above, one can also extend these bounded linear maps to AΓ. In fact,for each finite X ⊂ Γ and any n ≥ 0 the maps ∆X(n) : AΓ → Aloc

Γ are defined by

(4.16) ∆X(0) = ΠX and ∆X(n) = ΠX(n) −ΠX(n−1)

for any n ≥ 1. We note again that the range of ∆X(n) is contained in AX(n) regarded as a sub-

algebra of AlocΓ .

A typical use of the local decompositions is as follows. Fix Λ ⊂ P0(Γ), A ∈ AΛ, and X ⊂ Λ, anddenote by N the smallest integer n for which X(n) ∩Λ = Λ. Clearly, N depends on X and Λ, and∆Λ

X(n) = 0 for any n > N . Then, one can write

(4.17) A =∑

n≥0

∆ΛX(n)(A) =

N∑

n=0

∆ΛX(n)(A)

where this telescopic sum has terms with explicit, local support.For a quasi-local observable A ∈ AΓ and X ∈ P0(Γ), the conditional convergence of the infinite-

volume analog of (4.17), namely

(4.18) A =∑

n≥0

∆X(n)(A),

follows from noticing that ΠX(N)(A) =∑N

n=0∆X(n)(A), and invoking (4.13). In Section 5.1, we willdiscuss situations in which (4.18) converges absolutely. The remainder of this section is concernedwith continuity properties and basic estimates for the local approximations ΠX .

4.2.1. Continuity of local approximations. Given finite sets X ⊂ Λ ⊂ Γ, Proposition 4.3(ii) impliesthat the projection map ΠΛ

X preserves continuity in the strong operator topology. In particular, if

t 7→ A(t) ∈ AΛ is strongly continuous for all t in an interval I ⊆ R, then t 7→ ΠΛX(A(t)) ∈ AΛ is

also strongly continuous. In applications, we will be interested in a sequence of strongly continuousfunctions t 7→ AΛn(t) ∈ AΛn , with Λn ↑ Γ, that converges to a bounded map t 7→ A(t) ∈ AΓ.It will then be desirable that the localizing projections ΠY (A(t)), Y ∈ P0(Γ), also satisfy certaincontinuity properties.

While we do not have the standard von Neumann algebra setting where the notion of locallynormal is more natural, it is convenient to define a similar notion in our setting with C∗-algebraswithout reference to a representation.

Definition 4.6. A linear map K : AlocΓ → AΓ is called locally normal if there exists an increasing,

exhaustive sequence Λnn≥0 of finite subsets of Γ and corresponding bounded linear transforma-tions KΛn ∈ B(AΛn) with the following properties:

(i) For all n, KΛn : AΛn → AΛn is continuous on bounded subsets when both its domain andco-domain are considered with the strong operator topology;

(ii) Local uniform convergence of KΛn to K: For all X ⊂ Γ finite and any ǫ > 0, there exists Nsuch that for all n ≥ N we have

(4.19) ‖K(A) −KΛn(A)‖ ≤ ǫ‖A‖, for all A ∈ AX .

Note that local uniform convergence implies that KΛn converges strongly to K. However, sinceN is allowed to depend on X, this convergence is in general not uniform in P0(Γ). If HΛn is finite-dimensional, property (i) is automatically satisfied. Let us now consider an example satisfyingDefinition 4.6.

Example 4.7. Consider a quantum lattice system comprised of (Γ, d) and AΓ. Let F be an F -function on (Γ, d) and Φ ∈ BF be an interaction. The map K : Aloc

Γ → AΓ given by

(4.20) K(A) =∑

Z∈P0(Γ)

[Φ(Z), A] for any A ∈ AlocΓ


is locally normal in the sense of Definition 4.6. In fact, let Λnn≥0 be any sequence of non-empty,finite subsets of Γ that are increasing and exhaustive. For each n ≥ 0, define KΛn : AΛn → AΛn bysetting

(4.21) KΛn(A) =∑

Z⊂Λn

[Φ(Z), A] for any A ∈ AΛn .

Fix n ≥ 0, let X ⊂ Λn and A ∈ AX . One checks that

(4.22) KΛn(A) =∑

Z⊂Λn:

Z∩X 6=∅

[Φ(Z), A]

and therefore,

(4.23) ‖KΛn(A)‖ ≤∑

x∈X

∑

z∈Λn

∑

Z⊂Λn:x,z∈Z

‖[Φ(Z), A]‖ ≤ 2‖Φ‖F ‖F‖|X|‖A‖

holds for any A ∈ AX , where we have used that Φ ∈ BF . Taking X = Λn, one sees that KΛn ∈B(AΛn). With n ≥ 0 fixed again, let Z ⊂ Λn. It is clear that A 7→ [Φ(Z), A] satisfies Definition 4.6(i) on AΛn. As a finite sum of such terms, it is clear that KΛn satisfies Definition 4.6 (i) as well.Lastly, let X ∈ P0(Γ), A ∈ AX , and N ≥ 1 be sufficiently large so that X ⊂ ΛN . Again, one checksthat for any n ≥ N

(4.24) K(A)−KΛn(A) =∑

Z∈P0(Γ):

Z∩X 6=∅,Z∩(Γ\Λn)6=∅

[Φ(Z), A]

and therefore,

(4.25) ‖K(A) −KΛn(A)‖ ≤ 2‖A‖‖Φ‖F∑

x∈X

∑

y∈Γ\Λn

F (d(x, y)) .

Since |X| <∞, Definition 4.6 (ii) holds as F is summable.

A simple consequence of Definition 4.6 (i) is the following: for each n ≥ 0, t 7→ KΛn(A(t)) isstrongly continuous if t 7→ A(t) is strongly continuous. The next lemma establishes that the sameproperty holds for the composition ΠY (K(A(t))) for any Y ∈ P0(Γ).

Lemma 4.8. Let X,Y ∈ P0(Γ), ΠY : AΓ → AY be the extension of the map defined in (4.11), andK : Aloc

Γ → AΓ be a locally normal map. Then, for every strongly continuous map t 7→ A(t) ∈ AX

defined on an interval I ⊂ R, the function t 7→ ΠY (K(A(t))) ∈ AY , is also strongly continuous.

Proof. For ψ ∈ HY , define f(t) = ΠY (K(A(t)))ψ. We will prove that f : I → HY is continuousby showing that on compact intervals it is the uniform limit of a sequence of continuous functions.Let KΛn be a sequence of maps of the type described in Definition 4.6. For n large enough so that

X ⊂ Λn, define fn(t) = ΠΛn

Y (KΛn(A(t)))ψ. Since K is locally normal, each fn is continuous; here

we are using Propostion 4.3(i) and that t 7→ KΛn(A(t)) is strongly continuous by Definition 4.6(i).Using compatibility, see Proposition 4.5(ii), it is clear that fn(t) = ΠY (KΛn(A(t)))ψ. Now for anycompact set J ⊂ I, the estimate

supt∈J

‖fn(t)− f(t)‖ ≤ ‖ΠY ‖ supt∈J

‖KΛn(A(t)) −K(A(t))‖‖ψ‖

≤ ǫ‖ψ‖ supt∈J

‖A(t)‖,(4.26)

follows from ‖ΠY ‖ = 1 and local uniform convergence. Since A(t) is locally bounded and J iscompact, this proves the claim.


Two comments are in order. First, by Proposition 4.3(ii), for any Z ∈ P0(Γ) such that Y ⊂ Z,t 7→ ΠY (K(A(t))) considered as a map into AZ is also strongly continuous. Second, if t 7→ A(t)is, in fact, continuous in the norm topology on AX (in particular, if dimHX < ∞), and K isbounded, the result of the Lemma 4.8 is trivial since the bounded linear map ΠY K preserves thenorm-continuity.

We will also encounter one-parameter families of locally normal maps, Ks | s ∈ I, that arestrongly continuous in s and uniformly locally normal in the sense of the following definition.

Definition 4.9. Let I ⊂ R be an interval. A family of linear maps Ks : AlocΓ → AΓ, s ∈ I, is called a

strongly continuous family of uniformly locally normal maps if there exists an increasing, exhaustivesequence Λnn≥0 of finite subsets of Γ and families of bounded linear maps KΛn

s ∈ B(AΛn) stronglycontinuous in s, with the following properties:

(i) For all n and s, KΛns : AΛn → AΛn is continuous on bounded subsets when both its domain

and co-domain are considered with the strong operator topology, and this continuity isuniform for s ∈ I.

(ii) Uniform local convergence of KΛns to Ks: For all X ⊂ Γ finite and any ǫ > 0, there exists N

such that for all n ≥ N we have

(4.27) ‖Ks(A)−KΛns (A)‖ ≤ ǫ‖A‖, for all A ∈ AX , and all s ∈ I.

In (i), uniform for s ∈ I, means that given a bounded net Aαα∈I converging strongly to A andǫ > 0, there exists a choice of α0 ∈ I, independent of s, so that

‖KΛns (Aα)ψ −KΛn

s (A)ψ‖ < ǫ, for all α ≥ α0.

For families Ks, s ∈ I with I an infinite interval, the uniformity asked for in part (ii) of thisdefinition will typically not hold and one is led to consider subfamilies parametrized by s ∈ I0 ⊂ I,for compact intervals I0. Also note that the properties of a strongly continuous family of uniformlylocally normal maps imply that s→ Ks is strongly continuous by the usual ǫ/3 argument. We havenot assumed, however, that the maps Ks are bounded. In general, Ks is only locally bounded andcannot be extended to all of AΓ.

We now discuss two examples. The first is for a model with uniformly bounded on-sites, whilethe second does not require this assumption.

Example 4.10. Consider a quantum lattice system comprised of (Γ, d) and AΓ. Let F be an F -function on (Γ, d), I ⊂ R be an interval, and Φ ∈ BF (I) be a strongly continuous interaction. Foreach s0 ∈ I and any compact I0 ⊂ I, we claim that Kt : Aloc

Γ → AΓ given by

(4.28) Kt(A) = τt,s0(A) for any A ∈ AlocΓ and t ∈ I0 ,

is a strongly continuous family of uniformly locally normal maps in the sense of Definition 4.9.Here, the dynamics in (4.28) we are using is the infinite volume dynamics corresponding to Φ,from Theorem 3.5, with Hz = 0 for all z ∈ Γ.

To see that this is an example of Definition 4.9, let Λnn≥0 be any non-empty sequence offinite subsets of Γ that are increasing and exhaustive. For each n ≥ 0 and any t ∈ I0, defineKΛn

t : AΛn → AΛn by setting

(4.29) KΛnt (A) = τΛn

t,s0(A) for any A ∈ AΛn ,

the finite-volume dynamics associated to Φ.Fix n ≥ 0, t0 ∈ I0, and A ∈ AΛn. It is clear that

(4.30) KΛnt (A)−KΛn

t0 (A) =

∫ t

t0

τΛnr,s0([iH

ΦΛn

(r), A]) dr


holds, in the strong sense, for any t ∈ I0. Thus

(4.31) ‖KΛnt (A)−KΛn

t0 (A)‖ ≤ 2‖F‖|Λn|‖A‖∫ max(t0,t)

min(t0,t)‖Φ‖F (r) dr

and therefore, KΛnt ∈ B(AΛn) is strongly continuous in t for each n ≥ 0. Here we have argued as

in the proof of (8.47) in Corollary 8.5 using that ‖Φ‖F is locally integrable.For each n ≥ 0, one can show that property (i) of Definition 4.9 holds by arguing as in the proof

of Proposition 2.7(iii), and using that I0 is compact and ‖Φ‖F is locally bounded.Finally, we observe that (ii) is a simple consequence of Corollary 3.6 (iii).

Example 4.11. Consider a quantum lattice system comprised of (Γ, d) and AΓ. Fix a collectionof densely defined, self-adjoint on-site Hamiltonians Hzz∈Γ. Let F be an F -function on (Γ, d),I ⊂ R be an interval, and take Φs ∈ BF (R) for each s ∈ I. In this case, for any w ∈ L1(R) thefamily Kss∈I of linear maps with Ks : Aloc

Γ → AΓ given by

(4.32) Ks(A) =

∫

R

τ st (A)w(t) dt for all A ∈ AlocΓ and s ∈ I

is well-defined. Here for each fixed s ∈ I, τ st is the infinite-volume dynamics corresponding to Φs

whose existence is proven in Theorem 3.5.We will show that, under some additional assumptions on Φs, for each compact I0 ⊂ I, Kss∈I0

is a strongly continuous family of uniformly locally normal maps in the sense of Definition 4.9.These assumptions are:

(i) For each Z ∈ P0(Γ), (s, t) 7→ Φs(Z, t) is jointly strongly continuous on I0 × R.(ii) For each Z ∈ P0(Γ) and t ∈ R, s 7→ Φs(Z, t) is strongly differentiable, and its derivative

(s, t) 7→ Φ′s(Z, t) is jointly strongly continuous on I0 × R.

(iii) For each s ∈ I0, Φ′s ∈ BF (R) and moreover, for each T > 0

(4.33) sups∈I0

∫ T

−T‖Φs‖F (t) dt <∞, sup

s∈I0

∫ T

−T‖Φ′

s‖F (t) dt <∞.

To prove the above claim, choose any sequence Λnn≥0 of non-empty, increasing, exhaustivefinite subsets of Γ. For each such n ≥ 0 and any s ∈ I0, define approximating maps KΛn

s : AΛn →AΛn by setting

(4.34) KΛns (A) =

∫

R

τΛn,st (A)w(t) dt for all A ∈ AΛn and s ∈ I0 .

Here, τΛn,st (A) = U s

Λn(t, 0)∗AU s

Λn(t, 0) is the dynamics generated by the finite volume Hamiltonian

(4.35) HsΛn

(t) =∑

z∈Λn

Hz +∑

Z⊂Λn

Φs(Z, t).

We first show that for each n ≥ 0, the map KΛns ∈ B(AΛn) is strongly continuous. In fact, the

argument below demonstrates that KΛns is uniformly continuous in the operator norm on B(AΛn).

Fix n ≥ 0. Let s ∈ I0 and A ∈ AΛn. Assumptions (i) and (ii) above guarantee that the strongderivative of the finite-volume dynamics satisfies

(4.36)d

dsτΛn,st (A) = i

∑

Z⊂Λn

∫ t

0τΛn,sr ([Φ′

s(Z, r), τΛn,st,r (A)]) dr .

Using Assumption (iii) and the estimate (3.66), for any X ⊂ Λn, each A ∈ AX, and any T > 0,there exists M > 0 such that

(4.37) sups∈I0

supt∈[−T,T ]

∥

∥

∥

∥

d

dsτΛn,st (A)

∥

∥

∥

∥

≤ 2‖A‖‖F‖|X|Me2CF M


Now, let s0 ∈ I0 and take ǫ > 0. Since w ∈ L1(R), it is clear that there exists T > 0 for which

(4.38)

∫

|t|>T|w(t)| dt ≤ ǫ

4.

A short calculation shows that

(4.39)

∥

∥

∥

∥

∥

∫

|t|≤T

(

τΛn,st (A)− τΛn,s0

t (A))

w(t) dt

∥

∥

∥

∥

∥

≤ 2‖A‖‖F‖|X|Me2CF M‖w‖1|s− s0|

and thus for s sufficiently close to s0,

(4.40) ‖KΛns (A)−KΛn

s0 (A)‖ ≤ ǫ‖A‖.This proves the claimed continuity of KΛn

s as a function of s.We now prove (i). Again fix n ≥ 0. Let Aαα∈I be a bounded net in AΛn which converges in

the strong operator topology to A. Denote by B = supα∈I ‖Aα‖ < ∞, and note that ‖A‖ ≤ B. Letǫ > 0. Take T > 0 as in (4.38) and define δ > 0 by requiring that

(4.41) 4‖F‖|Λn|Me2CFM‖w‖1δ ≤ ǫ

For this choice of δ > 0, compactness of I0 implies that there is some N ≥ 1 and numberss1, s2, · · · , sN ∈ I0 for which the balls of radius δ centered at sj (1 ≤ j ≤ N) cover I0.

For each 1 ≤ j ≤ N , it is clear that KΛnsj (Aα) → KΛn

sj (A) in the strong operator topology. In this

case, for any ψ ∈ HΛn and ǫ as above, there is an α0 ∈ I for which

(4.42)∥

∥

∥

(

KΛnsj (Aα)−KΛn

sj (A))

ψ∥

∥

∥ ≤ ǫB‖ψ‖for all 1 ≤ j ≤ N whenever α ≥ α0.

In this case, for any s ∈ I0 there is a value of j for which∥

∥

(

KΛns (Aα)−KΛn

s (A))

ψ∥

∥ ≤∥

∥

∥

(

KΛns (Aα)−KΛn

sj (Aα))

ψ∥

∥

∥

+∥

∥

∥

(

KΛnsj (Aα)−KΛn

sj (A))

ψ∥

∥

∥

+∥

∥

∥

(

KΛnsj (A)−KΛn

s (A))

ψ∥

∥

∥

≤ 3ǫB‖ψ‖(4.43)

and we have proven (i).Lastly, we need to verify uniform local convergence of KΛn

s to Ks. Fix X ∈ P0(Γ) and A ∈ AX .Let ǫ > 0. Choose T > 0 as in (4.38). For any n ≥ 0 such that X ⊂ Λn, Corollary 3.6 (iii) implies

(4.44) ‖τ st (A)− τΛn,st (A)‖ ≤ 2‖A‖

CFe2It,0(Φs)It,0(Φs)

∑

x∈X

∑

y∈Γ\Λn

F (d(x, y)).

Since F is summable, for each x ∈ X there exists Λx ∈ P0(Γ) such that

(4.45)∑

y∈Γ\Λx

F (d(x, y)) ≤ ǫe−2CFM

4|X|‖w‖1M.

For n ≥ 0 sufficiently large that X ⊂ Λn and⋃

x∈X Λx ⊂ Λn, one has that

‖Ks(A)−KΛns (A)‖ ≤

∫

|t|≤T‖τ st (A)− τΛn,s

t (A)‖ |w(t)| dt + 2‖A‖∫

|t|>T|w(t)| dt

≤ 2‖w‖1‖A‖e2CFMM∑

x∈X

∑

y∈Γ\Λn

F (d(x, y)) + ǫ‖A‖/2

≤ ǫ‖A‖(4.46)


from which the claim is proved.

Our next result shows that given a strongly continuous family of uniformly locally normal mapsKs and any Y ∈ P0(Γ), the map (s, t) 7→ ΠY (Ks(A(t))), is jointly strongly continuous whenevert 7→ A(t) is strongly continuous. In particular, we have continuity on the diagonal t = s. Theresult, of course, also applies to the finite-volume setting where we can take Y = Λ = Γ.

Lemma 4.12. Let I and J be intervals, X,Y ∈ P0(Γ), ΠY : AΓ → AY be the extension of themap defined in (4.11), and Ks : Aloc

Γ → AΓ, s ∈ I be a strongly continuous family of uniformlylocally normal maps. Then, for every strongly continuous t 7→ A(t) ∈ AX , t ∈ J , the function(s, t) 7→ ΠY (Ks(A(t))) ∈ AY is jointly strongly continuous in t and s.

Proof. By the assumptions, there exists an increasing, exhaustive sequence Λnn≥0 of finite subsetsof Γ and bounded linear transformations KΛn

s ∈ B(AΛn), strongly continuous in s, that approximateKs as in Definition 4.9.

Let X,Y ∈ P0(Γ), ψ ∈ HY , and t → A(t) ∈ AX be strongly continuous. Define f(s, t) =ΠY (Ks(A(t)))ψ ∈ HY . We prove that f(s, t) is jointly continuous. Without loss of generality, wemay assume that I is compact. Fix (s0, t0). Since

‖f(s, t)− f(s0, t0)‖ ≤ ‖f(s, t)− f(s, t0)‖+ ‖f(s, t0)− f(s0, t0)‖,the joint continuity of f(t, s) can be obtained by proving the following two properties:

(a) f(s, t0) is continuous in s.(b) f(s, t) is an equicontinuous family of functions of t parameterized by s ∈ I.For (a), let ǫ > 0. Using Definition 4.9 (ii), pick n so that

‖Ks(A(t0))−KΛns (A(t0))‖ ≤ ǫ for all s ∈ I.

Using that ‖ΠY ‖ = 1, the continuity of f(·, t0) follows from the strong continuity of KΛns as

‖f(s, t0)− f(s0, t0)‖ ≤ 2ǫ‖ψ‖ + ‖KΛns (A(t0))−KΛn

s0 (A(t0))‖‖ψ‖.For (b), let ǫ > 0. We show that there exists δ > 0, such that |t− t0| < δ implies

(4.47) ‖f(s, t)− f(s, t0)‖ ≤ ǫ, for all s ∈ I.To see this, let fn(s, t) = ΠY (KΛn

s (A(t)))ψ, and J be a compact interval containing a neighborhoodof t0. Since ‖A(t)‖ is uniformly bounded for t ∈ J , using Definition 4.9(ii), choose n so that

(4.48) supt∈J

‖fn(s, t)− f(s, t)‖ ≤ ǫ, for all s ∈ I.

Now consider the family of functions fn(s, t) parameterized by s ∈ I. By Definition 4.9(i), since‖A(t)‖ is bounded on J , KΛn

s (A(t)) is strongly continuous in t . Since sups∈I ‖KΛns ‖ < ∞ by the

Uniform Boundedness Principle, it follows that ‖KΛns (A(t))‖ is bounded on I × J , and so

ΠY (KΛns (A(t))) = ΠΛn∪Y

Y (KΛns (A(t)))

is strongly continuous in t by Proposition 4.3. The argument used in Proposition 4.3 shows thatthe strong continuity of ΠY (KΛn

s (A(t))) is uniform in s ∈ I. In particular, there is a δ > 0 suchthat |t− t0| < δ implies

(4.49) ‖fn(s, t)− fn(s, t0)‖ < ǫ for all s ∈ I.

The equicontinuity of f(s, t) follows from (4.48) and (4.49).

In all the proofs above, we have used results on the finite volume local approximates to obtainresults for the infinite volume local approximates. However, there may be instances where onewants to work in a suitable representation of the infinite-volume algebra. We conclude this sectionwith a result regarding the GNS representation of a locally normal state.


Proposition 4.13. Let HΛ,Λ ∈ P0(Γ), be the family Hilbert spaces defined in (3.1), ω a locallynormal state on Aloc

Γ , and πω : AlocΓ → B(Hω) its corresponding GNS representation. The map

πω|AΛ: B(HΛ) → B(Hω) is continuous on arbitrary bounded subsets of its domain with respect to

the strong operator topology on both its domain and codomain.

Proof. Let Aα | α ∈ I be a net in the unit ball of B(HΛ) that converges to A in the strongoperator topology. To prove the claim, it is sufficient to verify that πω(A − Aα)ψ → 0, for all ψin the dense subspace of Hω given by vectors of the form πω(B)Ωω, where B ∈ Aloc

Γ and Ωω is thecyclic vector representing ω. To this end, note that

‖πω((A−Aα)B)Ωω‖2 = ω(B∗(A−Aα)∗(A−Aα)B).

Since A and the Aα belong to AΛ and B ∈ AlocΓ , there exists Λ′ ∈ P0(Γ) such that B∗(A−Aα)

∗(A−Aα)B ∈ AΛ′ . Since ω is locally normal, its restriction to AΛ′ is given by a density matrix ρΛ′ onHΛ′ . By writing ρΛ′ =

∑

n≥1 ρn|ξn〉〈ξn| in terms of an orthonormal set of eigenvectors ξn ∈ HΛ′ , itfollows that

‖πω(A−Aα)πω(B)Ωω‖2 = TrρΛ′B∗(A−Aα)∗(A−Aα)B =

∑

n≥1

ρn‖(A−Aα)Bξn‖2.

The result follows from using the analogous arguments from the proof of Proposition 4.3.

5. Quasi-local maps

In Section 3, we proved a Lieb-Robinson bound for the dynamics associated with a sufficientlylocal interaction. In addition to estimating the speed of propagation of a dynamically evolvedobservable, these bounds imply that the dynamics for a fixed time t is quasi-local. As a result,they can be well approximated by local observables as shown in Corollary 4.4. In recent years,other quasi-local maps have played a key role in proving both locality estimates of the spectralflow [12,54,59,93] and spectral gap stability of frustration-free quantum lattice models [21,22,56,83, 85, 96, 97, 130]. While we will consider both of these topics, the former in Section 6 and thelatter in [96], the focus of this section is the general study of quasi-local maps, see (5.1) below, andthe investigation of the key properties that will be useful in above mentioned applications. Thereexists a broad range of other applications that we will not discuss here [5–9,46].

We begin by showing how to apply the techniques from Section 4 to obtain estimable localapproximations of quasi-local maps. In Section 5.2, we provide a number of examples that willarise in our applications, including the difference of two dynamics. We discuss compositions of twoquasi-local maps in Section 5.3, and prove sufficient conditions for which the composition is againquasi-local. In Section 5.4 consider the composition of a quasi-local map with an interaction. Weshow that under suitable conditions such a composition can be rewritten as a local interaction.Moreover, we quantify the decay of the resulting interaction in terms of the decays of the originalinteraction and the quasi-local map. If the transformed interaction has sufficient decay, then thetheory developed in Section 3 applies and an infinite volume dynamics exists. We conclude inSection 5.5 by returning to the example of the difference of two dynamics and proving a continuityresult.

5.1. General quasi-local maps. Let (Γ, d), and AΓ be a quantum lattice system defined as inSection 3.1.1. A linear map K : Aloc

Γ → AΓ is said to satisfy a quasi-locality bound of order q ≥ 0if there is C < ∞ and a non-increasing function G : [0,∞) → [0,∞) with limr→∞G(r) = 0, suchthat for all X,Y ∈ P0(Γ), and A ∈ AX , B ∈ AY

(5.1) ‖[K(A), B]‖ ≤ C|X|q‖A‖‖B‖G(d(X,Y )).

Any linear mapping K satisfying (5.1) will be referred to as quasi-local. When relevant, we willdenote by CK(q,G) the smallest constant for which (5.1) holds. Since the function G in (5.1) abovegoverns the decay of the quasi-local map, we may refer to G as a decay function associated to K.


In this work, we will always assume that quasi-local maps are linear. However, there may be othercontexts in which it might be appropriate to generalize this definition.

The dependence of the bound in (5.1) on the support of the observable A through the factor|X|q is a choice we made based on the applications we have in mind. However, under appropriateassumptions, most of the estimates proved in this section also hold for quasi-local maps with moregeneral functions of |X|.

In most of our applications, the metric space (Γ, d) is equipped with an F -function F . In thiscase, one can often estimate a quasi-local map K as follows: there is C < ∞ such that for allX,Y ∈ P0(Γ), any A ∈ AX , and B ∈ AY we have

(5.2) ‖[K(A), B]‖ ≤ C‖A‖‖B‖GF (X,Y ) where GF (X,Y ) =∑

x∈X

∑

y∈Y

F (d(x, y)).

As we will see below, in certain estimates the bound (5.2) has advantages over (5.1). A simpleover-counting argument shows that

GF (X,Y ) ≤ |X|G(d(X,Y )), where G(r) := supx∈Γ

∑

y∈Γ:d(x,y)≥r

F (d(x, y)).

It follows from the uniform integrability of F , see (3.8), that G(r) → 0 as r → ∞. When thecorresponding F -function is weighted, i.e. F = Fg as defined (3.11), one has that

(5.3) GFg(X,Y ) =∑

x∈X

∑

y∈Y

Fg(d(x, y)) =∑

x∈X

∑

y∈Y

e−g(d(x,y))F (d(x, y)) ≤ ‖F‖|X|e−g(d(X,Y ))

and so, in this case, an estimate of the form (5.2) reduces to that of (5.1). For more detailedinformation on F -functions, including weighted F -functions, see Sections 8.1-8.3.

We now demonstrate an important estimate concerning quasi-local maps. For this result, we usethe concepts introduced in Section 4.2 and in particular the localizing maps ΠX and ∆X(n), definedwith respect to a locally normal product state ρ, as in (4.11) and (4.16), respectively.

Lemma 5.1. Let (Γ, d) and AΓ be a quantum lattice system, and ρ a locally normal product stateon AΓ. Let K : Aloc

Γ → AΓ be a quasi-local map. For any X ∈ P0(Γ) and n ≥ 0,

(5.4) ‖K(A) −ΠX(n)(K(A))‖ ≤ 2C|X|q‖A‖G(n) for all A ∈ AX .

In particular, if the decay function associated to K is summable, i.e.∑

n≥0G(n) < ∞, then for

any X ∈ P0(Γ) and each A ∈ AX ,

(5.5) K(A) =∞∑

n=0

∆X(n)(K(A)) .

The series on the right-hand-side above is absolutely convergent in norm with a bound that isuniform in the choice of locally normal product state ρ.

Note that the result above, of course, also applies to finite systems. In particular, for any finiteΛ, the result holds for any quasi local map K : AΛ → AΛ.

Proof. To see that (5.4) is true, fix X ∈ P0(Γ), A ∈ AX , and n ≥ 0. Observe that for anyB ∈ Aloc

Γ\X(n), the estimate

(5.6) ‖[K(A), B]‖ ≤ C|X|q‖A‖‖B‖G(d(X, supp(B))).

follows from (5.1). In this case, an application of Corollary 4.4 with ǫ = C|X|q‖A‖G(n) implies(5.4) as claimed.

Recalling (4.16), for any integer n ≥ 1, one can write

(5.7) ∆X(n)(K(A)) =(

ΠX(n)(K(A)) −K(A))

−(

ΠX(n−1)(K(A)) −K(A))


and therefore, an immediate consequence of (5.4) is the estimate

(5.8) ‖∆X(n)(K(A))‖ ≤ 4C|X|q‖A‖G(n − 1),

which is valid for any X ∈ P0(Γ), A ∈ AX , and n ≥ 1.Since G(n) → 0 as n→ ∞, (5.4) implies

(5.9) limn→∞

‖K(A) −ΠX(n)(K(A))‖ = 0.

It is clear from (4.16) that the local decompositions have telescopic sums, i.e. for any N ≥ 1,

(5.10)N∑

n=0

∆X(n)(K(A)) = ΠX(N)(K(A)),

and thus the series on the right-hand-side of (5.5) is norm convergent. Note also that

(5.11)

∞∑

n=0

‖∆X(n)(K(A))‖ ≤ ‖K(A)‖ + 4C|X|q‖A‖∞∑

n=0

G(n),

and so the series is also absolutely convergent, with norm bound independent of ρ, as claimed.

If in Lemma 5.1 we had assumed K satisfies (5.2) instead of (5.1), (5.4) would become

(5.12) ‖K(A) −ΠX(n)(K(A))‖ ≤ 2C‖A‖GF (X,Γ \X(n))

where the right-hand-side above is finite and non-increasing (in n) by the uniform summability ofthe F -function F .

5.2. Examples of quasi-local maps. In this section, we discuss a few of the most commonexamples of quasi-local maps (defined as in (5.1)). In applications, quasi-local maps are oftenconstructed as the thermodynamic limit of appropriate finite-volume maps. We first describe thisclass of quasi-local maps as a general example. Each of the more concrete examples we presentlater in this section will be of this general form.

Example 5.2 (A General Example). Let q ≥ 0, C < ∞, and G be a non-increasing functionG : [0,∞) → [0,∞) with limr→∞G(r) = 0. Let Λn∞n=1 be a sequence of increasing and exhaustivefinite subsets of Γ. In particular, this means that Λn ⊂ Λn+1 for all n ≥ 1, and given any x ∈ Γ,there exists N ≥ 1 for which x ∈ ΛN . Suppose that for each n ≥ 1, there is a linear mapKn : AΛn → AΛn for which:

(i) Given any sets X,Y ⊂ Λn the bound

(5.13) ‖[Kn(A), B]‖ ≤ C|X|q‖A‖‖B‖G(d(X,Y ))

holds for all observables A ∈ AX and B ∈ AY .(ii) For each finite X ⊂ Γ and any ǫ > 0, there is an N ≥ 1 for which

(5.14) ‖Kn(A)−Km(A)‖ ≤ ǫ‖A‖ for any n,m ≥ N and all A ∈ AX .

In this case, the local Cauchy assumption in part (ii) above, implies that a linear map K : AlocΓ → AΓ

is well-defined by setting

(5.15) K(A) = limn→∞

Kn(A) for each A ∈ AlocΓ ,

with the limit above being in norm. Moreover, for any finite sets X,Y ⊂ Γ, there is N ≥ 1 largeenough so that X ∪Y ⊂ ΛN since the sequence of sets is exhaustive. In this case, for any A ∈ AX ,B ∈ AY and n ≥ N ,

‖[K(A), B]‖ ≤ ‖[Kn(A), B]‖ + ‖[(K(A) −Kn(A)) , B]‖≤ C|X|q‖A‖‖B‖G(d(X,Y )) + 2‖B‖‖K(A) −Kn(A)‖.(5.16)


Here we have used the uniform quasi-locality estimate in (i), see (5.13). As the final term abovevanishes in the limit as n → ∞, it is clear that the map K defined in (5.15) above is quasi-local,see (5.1), and satisfies the same uniform quasi-local estimates as the collection Knn≥1.

For later applications, we now describe a variant of the estimate in Lemma 5.1 which is partic-ularly relevant to quasi-local maps of the type from Example 5.2.

Corollary 5.3. Let ρ be a locally normal product state on AΓ, and suppose that K and Kn∞n=1are maps that satisfy the uniform quasi-local and local Cauchy conditions from (5.13) and (5.14).For any X ∈ P0(Γ), m ≥ 1, and n ≥ 1 large enough so that X(m) ⊂ Λn, the bound

(5.17) ‖∆X(m)(K(A)) −∆Λn

X(m)(Kn(A))‖ ≤ min 2‖K(A) −Kn(A)‖, 8C|X|q‖A‖G(m − 1)

holds for any A ∈ AX where ∆Λn

X(m) : AΛn → AX(m) and ∆X(m) : AΓ → AX(m) are the local

decomposition maps defined in Section 4.2 (see (4.15) and (4.16)).

The bound above is particularly useful as it expresses the decay of the quantity on the left-hand-side above in both large n and m.

Proof. The proof uses two separate estimates. First, using consistency of the local decompositions,see Proposition 4.5(ii), it is clear that

(5.18) ∆X(m)(K(A)) −∆Λn

X(m)(Kn(A)) = ∆X(m)(K(A) −Kn(A))

and so the first part of the estimate holds since ΠX(m) is a norm one map. Note that this establishesthat the LHS of (5.17) decays in n. For the second part of the argument, we use that each termon the LHS of (5.17) can be estimated using Lemma 5.1. In fact,

‖∆X(m)(K(A)) −∆Λn

X(m)(Kn(A))‖ ≤ ‖∆X(m)(K(A))‖ + ‖∆Λn

X(m)(Kn(A))‖ ≤ 8C|X|q‖A‖G(m − 1)

as all maps considered satisfy the same quasi-local bound.

Example 5.4. Here we return to Example 4.7 and show that it has the form of the general examplediscussed in Example 5.2. Consider a quantum lattice system comprised of (Γ, d) and AΓ. Let F bean F -function on (Γ, d) and Φ ∈ BF be an interaction. As in (4.21), let Λnn≥0 be an increasing,exhaustive sequence of finite subsets of Γ and define Kn : AΛn → AΛn by setting

(5.19) Kn(A) =∑

Z⊂Λn

[Φ(Z), A] .

To see that (5.13) holds, let X,Y ∈ P0(Γ). For any A ∈ AX and n ≥ 0 large enough so thatX ⊂ Λn, one has that (4.22) holds and therefore,

(5.20) ‖Kn(A)‖ ≤∑

x∈X

∑

z∈Λn

∑

Z⊂Λn:x,z∈Z

‖[Φ(Z), A]‖ ≤ 2‖A‖‖Φ‖FGF (X,Λn)

where we use the notation GF of (5.2). As any F -function is uniformly integrable, the boundabove is uniform in the finite volume since GF (X,Λn) ≤ |X|‖F‖. For n ≥ 0 large enough so thatX ∪ Y ⊂ Λn, for each A ∈ AX , and any B ∈ AY , a simple commutator bound then yields:

(5.21) ‖[Kn(A), B]‖ ≤ 2‖Kn(A)‖‖B‖ ≤ 4‖Φ‖F ‖F‖|X|‖A‖‖B‖ .When X ∩ Y = ∅, a better estimate is achieved by observing that

(5.22) ‖[Kn(A), B]‖ ≤∑

Z⊂Λn:

Z∩X 6=∅,Z∩Y 6=∅

‖[[Φ(Z), A], B]‖ ≤ 4‖Φ‖F ‖A‖‖B‖GF (X,Y ).

This gives a quasi-locality estimate (uniform in the finite volume) of the type in (5.13).


To see (5.14), fix a finite set X ⊂ Γ, A ∈ AX , and take m ≤ n with m large enough so thatX ⊂ Λm. One checks that

(5.23) Kn(A) −Km(A) =∑

Z⊂Λn:

Z∩X 6=∅,Z∩(Λn\Λm)6=∅

[Φ(Z), A]

and therefore,

(5.24) ‖Kn(A)−Km(A)‖ ≤ 2‖A‖‖Φ‖FGF (X,Λn \ Λm).

Since F -functions are integrable, the locally-Cauchy estimate (5.14) follows.Under appropriate assumptions, one can generalize this example to prove quasi-locality estimates

for mappings of the form

(5.25) KΛ(A) =

N∑

j=1

CjADj with Cj ,Dj ∈ AΛ.

For example, the Lindblad generator of a quantum dynamical semigroup is of this form. See [98,105].

Example 5.5 (The Dynamics Associated to Φ ∈ BF (I)). Let (Γ, d) and AΓ be a quantum latticesystem, and I ⊂ R be an interval. Given an F -function, F , recall that a strongly continuousinteraction Φ : P0(Γ)× I → Aloc belongs to BF (I) if the function ‖Φ‖F : I → [0,∞) defined by

(5.26) ‖Φ‖F (t) = supx,y∈Γ

1

F (d(x, y))

∑

Z∈P0(Γ):

x,y∈Z

‖Φ(Z, t)‖

is locally bounded on I, see (3.18). Fix Φ ∈ BF (I) and let Hzz∈Γ be a collection of densely defined,self-adjoint on-site Hamiltonians. For any Λ ∈ P0(Γ), considered the finite-volume Hamiltonians

(5.27) H(Φ)Λ (t) =

∑

z∈Λ

Hz +∑

Z⊂Λ

Φ(Z, t) for any t ∈ I

and associated to them the dynamics τΛt,ss,t∈I , defined in (3.56). Theorem 3.3 demonstrates thatthese dynamics are quasi-local maps. In fact, this result shows that for any X,Y ⊂ Λ with X∩Y = ∅,

(5.28) ‖[τΛt,s(A), B]‖ ≤ 2‖A‖‖B‖CF

(

e2It,s(Φ) − 1)

GF (X,Y )

for all A ∈ AX , B ∈ AY , and s, t ∈ I. Here the number CF is the convolution constant associatedto the F -function F on Γ, see (3.9), and

(5.29) It,s(Φ) = CF

∫ max(t,s)

min(t,s)‖Φ‖F (r) dr .

Moreover, the bound proven in Theorem 3.4 (ii) shows that for any finite sets X ⊂ Λ0 ⊂ Λ,

(5.30) ‖τΛt,s(A) − τΛ0t,s (A)‖ ≤ 2C−1

F It,s(Φ)e2It,s(Φ)‖A‖GF (X,Λ \ Λ0)

holds for all A ∈ AX and s, t ∈ I. Again, this suffices to establish that these dynamics are locallyCauchy in the sense of (5.14), and so again this example is of the general form in Example 5.2.

Example 5.6 (The Difference of Two Dynamics). A quasi-local map that comes up in our appli-cations related to stability in [96] is the difference of two dynamics. More precisely, consider thesetting from Example 5.5. Fix a collection of densely defined, self-adjoint on-site HamiltoniansHzz∈Γ and two interactions Φ,Ψ ∈ BF (I). For any Λ ∈ P0(Γ), consider the Hamiltonians

(5.31) H(Φ)Λ (t) =

∑

z∈Λ

Hz +∑

Z⊂Λ


∑

z∈Λ

Hz +∑

Z⊂Λ

Ψ(Z, t)


and their corresponding dynamics, which we denote by τΛt,s and αΛt,s, respectively. For any s, t ∈ I,

a linear map KΛt,s : AΛ → AΛ is defined by

(5.32) KΛt,s(A) = τΛt,s(A)− αΛ

t,s(A).

Since both of the dynamics used in the definition of this map are automorphisms, it is clear that‖KΛ

t,s‖ ≤ 2. A different estimate is provided by the local norm bound in Theorem 3.4(i), namely

(5.33) ‖KΛt,s(A)‖ ≤ 2C−1

F It,s(Φ−Ψ)e2min(It,s(Φ),It,s(Ψ))‖A‖GF (X,Λ) ,

where It,s(Φ−Ψ) is defined as in (3.21). The above bound better reflects the fact that this differenceis small if either Φ is close to Ψ in BF (I), or |s − t| is small. The bound in Theorem 3.4(ii), seealso (5.30) above, allows one to establish that these maps are locally Cauchy in the sense of (5.14).In Section 5.5 below, we prove that these maps KΛ

t,s are uniformly quasi-local in the sense of (5.13)with constant pre-factors that once again decay if either Φ is close to Ψ in BF (I) or |s− t| is small;as is the case in (5.33).

Example 5.7 (Weighted Integrals of Quasi-local Maps). We briefly mention another interestingclass of examples, which includes the spectral flow introduced in Section 6.3. Let (Γ, d) and AΓ bea quantum lattice system, and µ a measure on R. Suppose that for each t ∈ R there is a quasi-localmap Kt : Aloc

Γ → AΓ for which the map K : AlocΓ → AΓ given by

(5.34) K(A) =

∫

R

Kt(A) dµ(t)

is well-defined. If the family Ktt∈R is sufficiently quasi-local and∫

|t|≥x dµ(t) decays sufficiently

fast as x→ ∞, then the mapping defined in (5.34) is quasi-local with explicit decay estimates.For example, let w ∈ L1(R), and Φ ∈ BFa(R) where we recall that Fa is a weighted F -function of

the form Fa(r) = e−arF (r) with a > 0. Let τΛt := τΛt,0 be the finite-volume dynamics given in (3.6)that is associated with the local Hamiltonians of the form

HΛ(t) =∑

X⊂Λ

Φ(X, t).

Then, using Lieb-Robinson bounds, one can see that the map K : AΛ → AΛ defined by

K(A) =

∫

R

τΛt (A)w(t)dt

satisfies the following bound: for all A ∈ AX , B ∈ AY with X ∪ Y ⊆ Λ and X ∩ Y = ∅,‖[K(A), B]‖ ≤ 2‖A‖‖B‖|X|G(d(X,Y )), where G is the decreasing function

G(r) =‖w‖L1‖F‖

CFa

e−ar/2 +

∫

|t|>r/2va

|w(t)|dt,

and va is the Lieb-Robinson velocity, see the discussion following Theorem 3.1. To see this, notethat for all T ∈ R,

‖[K(A), B]‖ ≤∫

|t|≤T‖[τΛt (A), B]‖|w(t)|dt +

∫

|t|>T‖[τΛt (A), B]‖|w(t)|dt.

With the choice of T = d(X,Y )/2va, the bound is attained by applying (3.26) to the integral over|t| ≤ T , and using the trivial bound ‖[τΛt (A), B]‖ ≤ 2‖A‖‖B‖ for the integral over |t| > T .


5.3. Compositions of quasi-local maps. In applications, we find it useful to recognize certainmappings as the composition of quasi-local maps. When Γ is finite, these compositions are well-defined and estimates, as indicated below, follow readily. For sets Γ of infinite cardinality, morecare must be taken when defining such compositions. This section discusses two classes of exampleswhere these compositions are well-defined and we describe the estimates that follow.

It will be convenient to make an additional assumption on the metric space (Γ, d). We saythat (Γ, d) is ν-regular if the cardinality of balls in Γ grows at most polynomially, i.e. there existnon-negative κ and ν for which

(5.35) supx∈Γ

|bx(n)| ≤ κnν for any n ≥ 1 .

More comments about ν-regular metric spaces (Γ, d) can be found in Appendix 8.1. Under thisassumption, given any X ∈ P0(Γ) and n > 0, the cardinality of X(n), the inflation of X defined in(4.14), satisfies the following rough estimate:

(5.36) |X(n)| ≤∑

x∈X

|bx(n)| ≤ κnν |X|.

Let (Γ, d) and AΓ be a quantum lattice system on a ν-regular metric space. We will say that alinear map K : Aloc

Γ → AΓ is locally bounded if there are non-negative numbers p and B for which

(5.37) ‖K(A)‖ ≤ B|X|p‖A‖ for all A ∈ AX with X ∈ P0(Γ) .

More general growth in X, i.e. the support of A, could be allowed, but the above moment conditioncovers all of the applications we have in mind. As discussed in Section 5.1, we say that a linear mapK : Aloc

Γ → AΓ is quasi-local if there are non-negative numbers q and C as well as a non-increasingfunction G, G : [0,∞) → [0,∞), with limr→∞G(r) = 0 for which given any X,Y ∈ P0(Γ),

(5.38) ‖[K(A), B]‖ ≤ C|X|q‖A‖‖B‖G(d(X,Y )) for all A ∈ AX and B ∈ AY .

We will refer to C, q, and G as the parameters of the quasi-local map.We first consider compositions of linear maps for the following situation. Suppose that K1 :

AlocΓ → AΓ is locally bounded and quasi-local in that it satisfies both (5.37) and (5.38). Further-

more, assume that K2 : AΓ → AΓ is linear, bounded, and quasi-local. So, in particular, there arenon-negative numbers q2 and C2 and a decay function G2 for which the analogue of (5.38) holdsfor K2. In many applications, the mapping K2 arises as the unique linear extension of a bounded,quasi-local map K2 : Aloc

Γ → AΓ. In this situation, we can define the composition K : AlocΓ → AΓ

in the usual way, i.e.

(5.39) K(A) = K2(K1(A)) for all A ∈ AlocΓ .

Moreover, any such map satisfies the following estimate.

Lemma 5.8. Let (Γ, d) be ν-regular, K1 : AlocΓ → AΓ be a locally bounded, quasi-local map, and

K2 : AΓ → AΓ be a bounded, quasi-local map. For i = 1 or 2, denote by Bi, Ci, pi, qi, Gi thecorresponding parameters from (5.37) and (5.38). Then, the following hold for the compositionK = K2 K1:

(i) K is locally bounded: for any A ∈ AX with X ∈ P0(Γ),

(5.40) ‖K(A)‖ ≤ B|X|p‖A‖,where one may take B = B1‖K2‖ and p = p1.

(ii) For any A ∈ AX and B ∈ AY where X,Y ∈ P0(Γ),

(5.41) ‖[K(A), B]‖ ≤ ‖A‖‖B‖min2B|X|p, C|X|qG(d(X,Y )),where the numbers B and p may be taken as in (5.40), one may take q = p1 + q2, and

(5.42) C = max(κq2B1C2, 4C1‖K2‖) and G(r) = (r/2)q2νG2(r/2) +G1(r/2).


We note that if the functionG described in (5.42) above is non-increasing and satisfies limr→∞G(r) =0, then the above estimates show that K is quasi-local.

Proof. To prove (i), note that given any X ∈ P0(Γ),

(5.43) ‖K(A)‖ ≤ ‖K2‖‖K1(A)‖ ≤ B1‖K2‖|X|p1‖A‖ for all A ∈ AX .

This proves (5.40).The proof of (ii) follows from two observations. First, the bound in (5.40) implies a rough

estimate on the commutator for any X,Y ∈ P0(Γ). In fact, whenever A ∈ AX and B ∈ AY , onehas

(5.44) ‖[K(A), B]‖ ≤ 2‖K(A)‖‖B‖ ≤ 2B|X|p‖A‖‖B‖.Next, we note that when d(X,Y ) > 0, we obtain better estimates. Under this additional con-

straint, set m = d(X,Y )/2. For any locally normal product state ρ on AΓ, the estimate

(5.45) ‖K1(A)−ΠX(m)(K1(A))‖ ≤ 2C1|X|p1‖A‖G1(m)

follows from an application of Lemma 5.1. Denoting by Am = ΠX(m)(K1(A)), we find that

(5.46) ‖[K(A), B]‖ ≤ ‖[K2(Am), B]‖ + ‖[K2(K1(A)−Am), B]‖.Since K2 is quasi-local, we have that

‖[K2(Am), B]‖ ≤ C2|X(m)|q2‖Am‖‖B‖G2(d(X(m), Y ))

≤ κq2B1C2‖A‖‖B‖|X|p1+q2(d(X,Y )/2)q2νG2(d(X,Y )/2)(5.47)

where we have used the local bound for K1, (5.36), and the fact that d(X,Y ) ≤ 2d(X(m), Y ). Thesecond term in (5.46) above satisfies

‖[K2(K1(A)−Am), B]‖ ≤ 2‖K2‖‖K1(A)−Am‖‖B‖≤ 4C1‖K2‖‖A‖‖B‖|X|p1G1(m).(5.48)

This completes the proof of (5.41).

For some of our applications, the estimates proven in Lemma 5.8 do not suffice. Briefly, someinformation is lost when estimating the outer mapping K2 with a rough norm bound. Due to this,we consider more general compositions in the proposition below. First, we introduce some notation.Let G : [0,∞) → [0,∞) and m ≥ 0, we say that G has a finite m-th moment if

(5.49)

∞∑

n=0

(1 + n)mG(n) <∞ .

Proposition 5.9. Let (Γ, d) be a ν-regular metric space. For i = 1, 2, let Ki : AlocΓ → AΓ be

locally bounded, quasi-local maps. Suppose that G1, the decay function in (5.38) associated to K1,has a finite νp2-th moment. Then, for any choice of locally normal product state ρ on AΓ, thecomposition Kρ : Aloc

Γ → AΓ given by

(5.50) Kρ(A) =∞∑

n=0

K2(∆ρX(n)(K1(A))) for all A ∈ Aloc

Γ with supp(A) = X

is well-defined. The series above is absolutely convergent and may be estimated uniformly in thechoice of locally normal product state ρ. In fact, the mapping Kρ is independent of the choice of ρ.

Proof. Fix a locally normal product state ρ on AΓ. Lemma 5.1 shows that for each X ∈ P0(Γ) andany A ∈ AX we have that

(5.51) K1(A) =∞∑

n=0

∆ρX(n)(K1(A))


and the series above is absolutely convergent. To obtain this series representation, it is only requiredthat the decay function associated to K1 is summable, see e.g. (5.11) which is independent of thechoice of ρ.

We now claim that under the additional finite moment condition, for each X ∈ P0(Γ) and anyA ∈ AX , the series defining Kρ(A) in (5.50) is also absolutely convergent. In fact, the bound

∞∑

n=0

‖K2(∆ρX(n)(K1(A)))‖ ≤ B2

∞∑

n=0

|X(n)|p2‖∆ρX(n)(K1(A))‖

≤ B1B2|X|p1+p2‖A‖+ 4C1B2κp2 |X|p2+q1‖A‖

∞∑

n=1

nνp2G1(n− 1),(5.52)

can be obtained as follows. For the first inequality above, we use that K2 is locally bounded. Forthe second inequality, we first partition the sum into n = 0 and n > 0. For n = 0, we use that∆X = ΠX and K1 is locally bounded. For n > 0, we apply (5.8) using the quasi-locality of K1, andinvoke (5.36).

Now, let ρ1 and ρ2 be any two locally normal product states on AΓ. We show that for each fixedX ∈ P0(Γ) and any ǫ > 0, one can estimate

(5.53) ‖Kρ1(A)−Kρ2(A)‖ ≤ ǫ‖A‖ for all A ∈ AX

and hence prove that the mapping Kρ is independent of the choice of ρ.By the absolute convergence proven in (5.52) and the finite moment condition, it is clear that

for any ǫ > 0 there is some N ≥ 1, independent of ρ, for which

(5.54) 4C1B2κp2 |X|p2+q1

∞∑

n=N

(1 + n)νp2G1(n) < ǫ/3.

For N as above, we write

Kρ1(A)−Kρ2(A) =

N∑

n=0

(

K2(∆ρ1X(n)(K1(A)))−K2(∆

ρ2X(n)(K1(A)))

)

+∞∑

n=N+1

(

K2(∆ρ1X(n)(K1(A))) −K2(∆

ρ2X(n)(K1(A)))

)

.(5.55)

Based on of N , it follows from (5.52) that

(5.56)

∥

∥

∥

∥

∥

∞∑

n=N+1

(

K2(∆ρ1X(n)(K1(A)))−K2(∆

ρ2X(n)(K1(A)))

)

∥

∥

∥

∥

∥

<2ǫ

3‖A‖.

Using linearity of K2 and the telescopic properties of the sums, see (5.10), we also have that∥

∥

∥

∥

∥

N∑

n=0

(

K2(∆ρ1X(n)(K1(A)))−K2(∆

ρ2X(n)(K1(A)))

)

∥

∥

∥

∥

∥

(5.57)

=∥

∥

∥K2

(

Πρ1X(N)(K1(A))−Πρ2

X(N)(K1(A)))∥

∥

∥

≤ B2|X(N)|p2‖Πρ1X(N)(K1(A))−Πρ2

X(N)(K1(A))‖≤ 4C1B2κ

p2 |X|p2+q1Nνp2G1(N)‖A‖<ǫ

3‖A‖(5.58)

where we have used the local bound for K2 and inserted (and removed) K1(A) to apply Lemma 5.1.The final bound above is clear from (5.54). The claim in (5.53) now follows.


From Proposition 5.9, we now have conditions under which there is a well-defined compositionof two locally bounded, quasi-local maps. The next lemma provides local bounds and quasi-localestimates for the resulting composition.

Lemma 5.10. Let (Γ, d) be a ν-regular metric space. For i = 1, 2, let Ki : AlocΓ → AΓ be locally

bound, quasi-local maps. Suppose that G1, the decay function in (5.38) associated to K1, has afinite νp2-th moment and let K : Aloc

Γ → AΓ denote the composition from (5.50).

(i) K is locally bounded: for any A ∈ AX with X ∈ P0(Γ)

(5.59) ‖K(A)‖ ≤ B|X|p‖A‖,where one may take p = p2 +maxp1, q1 and

(5.60) B = B2

(

B1 + 4C1κp2

∞∑

n=0

(1 + n)νp2G1(n)

)

.

(ii) For any A ∈ AX and B ∈ AY where X,Y ∈ P0(Γ), one has that

(5.61) ‖[K(A), B]‖ ≤ ‖A‖‖B‖min

2B|X|p, C|X|qG(d(X,Y ))

where one may take C = maxκq2B1C2, 8κp2C1B2, q = maxp1, q1+maxp2, q2, and

(5.62) G(r) = (r/2)q2νG2(r/2) +∞∑

n=⌊r/2⌋

(1 + n)νp2G1(n).

Again we stress that the bounds above demonstrate that the composition is quasi-local if thefunction G in (5.62) is non-increasing with limr→∞G(r) = 0.

Proof. The bound (5.59) is a consequence of (5.52) found in the proof of Proposition 5.9. To prove(5.61), we argue as in the proof of Lemma 5.8(ii). We need only consider the case when d(X,Y ) > 0,and therein we set m = ⌊d(X,Y )/2⌋. For A ∈ AX , we write

(5.63) K(A) =∞∑

n=0

K2(∆X(n)(K1(A))) = K2(ΠX(m)(K1(A))) +∞∑

n=m+1

K2(∆X(n)(K1(A))).

Here we have used an expansion as in (5.50), and the telescopic property (5.10). Moreover, we havedropped the dependence of ρ from the notation since (5.50) is invariant under the choice of locallynormal product state by Proposition 5.9. The estimate

(5.64) ‖[K(A), B]‖ ≤∥

∥

[

K2(ΠX(m)(K1(A))), B]∥

∥+

∞∑

n=m+1

∥

∥

[

K2

(

∆X(n)(K1(A)))

, B]∥

∥

readily follows.As K2 is quasi-local, it is clear that∥

∥

[

K2(ΠX(m)(K1(A))), B]∥

∥ ≤ C2|X(m)|q2‖K1(A)‖‖B‖G2(d(X(m), Y ))

≤ B1C2κq2 |X|p1+q2‖A‖‖B‖(d(X,Y )/2)q2νG2(d(X,Y )/2)(5.65)

To estimate the remaining term, for each n ≥ m+ 1 we find∥

∥

[

K2

(

∆X(n)(K1(A)))

, B]∥

∥ ≤ 2∥

∥K2

(

∆X(n)(K1(A)))∥

∥ ‖B‖≤ 2B2|X(n)|p2‖∆X(n)(K1(A))‖‖B‖≤ 8C1B2κ

p2 |X|p2+q1‖A‖‖B‖np2νG1(n− 1)(5.66)

where we have used (5.8). This proves (5.61).


5.4. Quasi-local transformations of interactions. Important applications of quasi-local mapsarise in the classification of gapped ground state phases [12, 14, 16, 99–101] and recent proofs ofstability of the spectral gap [21,22,83,85,96,97]. In these proofs, key insights come from analyzingthe composition of a quasi-local map with an interaction, K Φ : P0(Γ) → AΓ. It is importantto note that such maps are not necessarily interactions themselves, as the image lies in the quasi-local algebra, AΓ, rather than the algebra of local observables, Aloc

Γ . In our applications, theinteraction and quasi-local map often depend on an auxiliary parameter and we allow for this inour construction and results. In what follows, we provide a general framework under which onecan construct a bona fide interaction from such a composition and derive estimates that determineconditions under which these transformed interactions have a finite F -norm.

We begin with a general description of transformed interactions in Section 5.4.1. In Section 5.4.2,we prove estimates on these transformed interactions in finite volume. In Section 5.4.3, we giveconditions under which the finite-volume results proven in Section 5.4.2 extend to the thermody-namic limit. A concrete application of these results will be given in Section 6.5, where we show thatthe spectral flow automorphisms can be realized as the dynamics generated by a time-dependentinteraction with good decay properties.

5.4.1. Transformed finite-volume Hamiltonians. To investigate the spectral properties of a givenHamiltonian H, it is often convenient to work with a unitarily equivalent Hamiltonian H ′ = U∗HU .When the original Hamiltonian is a sum of local terms, the strict locality of these terms is typicallynot preserved under the mapping H 7→ H ′. In recent applications, most notably the proof ofstability, it is shown that locality based arguments, such as Lieb-Robinson bounds, still apply toH ′ if the automorphism implemented by the unitary U is sufficiently quasi-local.

In this section, we discuss this situation more generally. Specifically, we analyze the transfor-mation of a given interaction by a quasi-local map. Briefly, we first argue that the compositionof a quasi-local map with an interaction can, using the methods of Section 4.2, still be realized asan interaction with strictly local terms. Moreover, we show that the spatial decay associated tothis new interaction can be estimated in terms of the decays of the original interaction and thequasi-local map. Finally, we discuss quasi-locality estimates for the dynamics of this transformedinteraction.

To establish some notation, let us first consider a simple, time-independent case in finite volume.As before, fix a quantum lattice system comprised of (Γ, d) and AΓ. Let Φ be an interaction on AΓ

and recall that for any finite Λ ⊂ Γ, we denote by

(5.67) HΦΛ =

∑

X⊂Λ

Φ(X)

the finite-volume Hamiltonian generated by Φ. Our goal here is to analyze the transformation ofthis local Hamiltonian HΦ

Λ by a linear map K : AΛ → AΛ. In particular, we consider

(5.68) K(HΦΛ ) =

∑

X⊂Λ

K(Φ(X)) .

Generally, the map K will not preserve locality, and in such cases, each term in (5.68) will beglobal in the sense that supp(K(Φ(X))) = Λ for each X ⊂ Λ. For this reason, the sum on theright-hand-side of (5.68) does not represent an interaction in the sense defined in Section 3.1.

Using the methods of Section 4.2, one can rewrite the right-hand-side of (5.68) as a sum of strictlylocal terms. To see this, fix a locally normal product state ρ on AΓ. In this finite-volume context,we only use the restriction of ρ to AΛ and again refer to it as ρ. In terms of ρ, we have defined localdecompositions with respect to any X ⊂ Λ and each n ≥ 0 as the maps ∆Λ

X(n) : AΛ → AΛ given by

(4.15). Recall further that ∆ΛX(n) has range contained in AX(n)∩Λ (using again the identification of

the former as a sub-algebra of AΛ), and moreover, ‖∆ΛX(n)‖ ≤ 2. In this case, each term K(Φ(X))


appearing in (5.68) can be written as a finite telescopic sum as in (4.17) by

(5.69) K(Φ(X)) =∑

n≥0

∆ΛX(n)(K(Φ(X))).

For any Z ⊂ Λ, define

(5.70) ΨΛ(Z) =∑

n≥0

∑

X⊂Z:X(n)∩Λ=Z

∆ΛX(n)(K(Φ(X)))

with the understanding that empty sums are taken to be zero. By construction, ΨΛ(Z) ∈ AZ

and under the additional assumption that K(A)∗ = K(A∗) for all A ∈ AΛ, we see that ΨΛ is awell-defined (finite-volume) interaction in the sense of Section 3.1. Moreover,

(5.71) K(HΦΛ ) =

∑

X⊂Λ

K(Φ(X)) =∑

Z⊂Λ

ΨΛ(Z) = HΨΛΛ .

In words, using the notation from (5.67), the final equalities in (5.71) show that the transformedHamiltonian in (5.68) may be rewritten as the Hamiltonian generated by the interaction ΨΛ.

5.4.2. Finite-volume results. In this section, we give a finite-volume analysis of the transformedinteractions briefly introduced at the end of Section 5.4.1. In Section 5.4.3, we will discuss ap-propriate conditions under which these results will extend to the thermodynamic limit. For manyof our applications, both the interaction and the quasi-local map will be time-dependent. As aconsequence, we state and prove our estimates for families of interactions and quasi-local maps.

We make two useful observations in this section. First, we indicate a set of continuity assumptionsunder which a finite-volume transformed interaction corresponds to a well-defined dynamics. Theseassumptions will also guarantee that the interaction which generates this transformed interaction isstrongly continuous in the sense of Section 3.1.1. Next, we will show that certain decay assumptionson the initial interaction Φ and quasi-local map K lead to explicit estimates on the decay of theinteraction ΨΛ; here we are using the notation introduced at the end of Section 5.4.1. Technically,the continuity and decay assumptions are independent, however, in most applications, the modelswe consider satisfy both sets of assumptions simultaneously.

Let (Γ, d) and AΓ be a quantum lattice system, and I ⊂ R be an interval. We once again workwith strongly continuous interactions Φ : P0(Γ)× I → Aloc

Γ ; meaning that, for all X ∈ P0(Γ),

(i) Φ(X, t)∗ = Φ(X, t) ∈ AX for all t ∈ I.(ii) The map t 7→ Φ(X, t) is continuous in the strong operator topology on AX = B(HX).

For any Λ ∈ P0(Γ), we define a finite-volume, time-dependent Hamiltonian associated to Φ by

(5.72) HΦΛ (t) =

∑

X⊂Λ

Φ(X, t) for all t ∈ I .

From the assumptions, it is clear that HΦΛ is also pointwise self-adjoint and strongly continuous as

it is the finite sum of such terms.For the remainder of this subsection, we fix a finite volume Λ ∈ P0(Γ), and are interested in

studying time-dependent transformed finite-volume Hamiltonians analogous to those consider inSection 5.4.1. Specifically, given any family of linear maps Kt : AΛ → AΛt∈I , we consider the setof all operators of the form

(5.73) Kt(HΦΛ (t)) =

∑

X⊂Λ

Kt(Φ(X, t)),

and will refer to such collections as a finite-volume family of transformed interactions. Underassumptions which guarantee that t 7→ Kt(H

ΦΛ (t)) is pointwise self-adjoint and strongly continuous,

the methods of Section 2.2, see also Section 3.1.1, demonstrate that these transformed interactions


correspond to a dynamics. More precisely, Proposition 2.2 shows that for any s, t ∈ I the strongsolution UΛ(t, s) ∈ AΛ of

(5.74)d

dtUΛ(t, s) = −iKt(H

ΦΛ (t))UΛ(t, s), UΛ(s, s) = 1

defines a two-parameter family of unitaries, and thus a cocycle of automorphisms τΛt,s of AΛ with

(5.75) τΛt,s(A) = UΛ(t, s)∗AUΛ(t, s) for all A ∈ AΛ.

We refer to τΛt,s as the dynamics corresponding to the transformed Hamiltonian in (5.73).A main goal of this subsection is to establish assumptions under which the dynamics in (5.75)

satisfies a quasi-locality estimate, also known as a Lieb-Robinson bound, see Theorem 3.1. In orderto do so, we first re-write the family of transformed interactions in (5.73) as a sum of strictly localterms. Fix a locally normal product state ρ on AΓ. For any Z ⊂ Λ and each t ∈ I, set

(5.76) ΨΛ(Z, t) =∑

n≥0

∑

X⊂Z:X(n)∩Λ=Z

∆ΛX(n)(Kt(Φ(X, t)))

where, as in Section 5.4.1, we have made local decompositions of the global terms on the right-hand-side of (5.73); compare with (5.69) and (5.70). We stress that for all t ∈ I, we make localdecompositions with respect to the same locally normal product state ρ. As in (5.71), it is clearthat for each t ∈ I,

(5.77) Kt(HΦΛ (t)) =

∑

X⊂Λ

Kt(Φ(X, t)) =∑

Z⊂Λ

ΨΛ(Z, t) = HΨΛΛ (t) .

We now introduce a set of assumptions on the family of functions Kt : AΛ → AΛt∈I whichguarantee that: (i) the dynamics in (5.75) is well-defined and (ii) the mapping ΨΛ : P0(Λ)×I → AΛ

is a strongly continuous interaction in the sense of Section 3.1.1.

Assumption 5.11. We assume the collection of finite-volume linear maps Kt : AΛ → AΛt∈I , isa strongly continuous family of strongly continuous transformations that are compatible with theinvolution in the sense that:

(i) For each t ∈ I, Kt(A)∗ = Kt(A

∗) for all A ∈ AΛ.(ii) For each A ∈ AΛ, the function t 7→ Kt(A) is norm continuous.(iii) For each t ∈ I, the map Kt : AΛ → AΛ is continuous on bounded subsets when both its

domain and co-domain are equipped with the strong operator topology and moreover, thiscontinuity is uniform on compact subsets of I.

Assumption 5.11(i), together with Proposition 4.5(v), is used to ensure that the terms definedin (5.76) are point-wise self-adjoint. This is important in defining the unitary propagator, but itplays no role in establishing various continuity properties. Next, as is discussed in Section 4.2.1,Assumption 5.11 (ii) and (iii) guarantee that t 7→ Kt(H

ΦΛ (t)) is strongly continuous. As such,

the finite-volume dynamics associated to this transformed interaction, see (5.74) and (5.75), iswell-defined. In particular, this dynamics is independent of the choice of ρ. Note further thatAssumption 5.11 (ii) and (iii) also ensure that for each X ⊂ Λ, t 7→ Kt(Φ(X, t)) is stronglycontinuous. Given this, Proposition 4.3 shows that each of the finitely many terms on the right-hand-side of (5.76) is strongly continuous as well, and as a result, ΨΛ is a strongly continuousinteraction. The interaction ΨΛ, which does depend on the choice of ρ, will be useful in proving aquasi-locality bound on the finite-volume dynamics in (5.75).

The goal for the remainder of this section is to quantify a quasi-locality estimate for the finite-volume dynamics in (5.75) in terms of decay properties of the original interaction Φ and the finite-volume transformations Ktt∈I . For these results, we assume that (Γ, d) is ν-regular and equippedwith an F -function F .


Let us again fix an interval I ⊂ R and a finite-volume Λ ∈ P0(Γ). We make the following decayassumptions on a family of finite volume transformations.

Assumption 5.12. We assume that the family of finite-volume linear maps Kt : AΛ → AΛt∈I is atime-dependent family of locally bounded, quasi-local maps in the sense that:

(i) There is some p ≥ 0 and a measurable, locally bounded function B : I → [0,∞) so thatgiven any X ⊂ Λ,

(5.78) ‖Kt(A)‖ ≤ B(t)|X|p‖A‖ for all A ∈ AX and t ∈ I .

(ii) There is some q ≥ 0, a non-increasing function G : [0,∞) → [0,∞) with G(r) → 0 asr → ∞, and a measurable, locally bounded function C : I → [0,∞) for which given anysets X,Y ⊂ Λ, one has that

(5.79) ‖[Kt(A), B]‖ ≤ C(t)|X|q‖A‖‖B‖G(d(X,Y )) for all A ∈ AX , B ∈ AY , and t ∈ I .

For the initial interaction, we impose decay assumptions which compensate for the factors of |X|found in (5.78) and (5.79) above. More precisely, for any time-dependent interaction Φ and eachm ≥ 0, we define a new interaction Φm, which we call the m-th moment of Φ, with terms

(5.80) Φm(X, t) = |X|mΦ(X, t) for all X ∈ P0(Γ) and t ∈ I .To prove the result in Theorem 5.13 below, we will assume that the initial interaction Φ satisfiesΦm ∈ BF (I) for m = maxp, q with p and q as in (5.78) and (5.79), respectively. Recall that aninteraction Φ ∈ BF (I) if and only if Φ : P0(Γ)× I → Aloc

Γ is a strongly continuous interaction andthe map ‖Φ‖F : I → [0,∞) given by

(5.81) ‖Φ‖F (t) = supx,y∈Γ

1

F (d(x, y))

∑

X∈P0(Γ):

x,y∈X

‖Φ(X, t)‖,

is locally bounded. An immediate consequence of (5.81) is that for any finite volume Λ ∈ P0(Γ)and any pair x, y ∈ Λ, the bound

(5.82)∑

X⊂Λ:x,y∈X

‖Φ(X, t)‖ ≤ ‖Φ‖F (t)F (d(x, y))

holds for all t ∈ I. We refer to Section 3.1.1 for more details on BF (I).Finally, before we state our first result we review some notation. Recall that a non-negative

function G : [0,∞) → (0,∞) has a finite m-th moment if

(5.83)∞∑

n=0

(1 + n)mG(n) <∞ .

Note that, in this case, the tail of the series r 7→ ∑

n=⌊r⌋(1 + n)mG(n) is a non-negative, non-

increasing function for which

(5.84) limr→∞

∞∑

n=⌊r⌋

(1 + n)mG(n) = 0.

We state our basic estimate on these finite-volume transformed interactions. In the statement,we make use of the quantities p, q and G from Assumption 5.12.

Theorem 5.13. Consider a quantum lattice system comprised of ν-regular metric space (Γ, d)and quasi-local algebra AΓ. Let F be an F -function on (Γ, d), I ⊂ R be an interval, and Λ ∈P0(Γ). Assume that Kt : AΛ → AΛt∈I is a quasi-local family of transformations satisfyingAssumption 5.12, and Φ is a strongly continuous interaction such that Φm ∈ BF (I) for m =maxp, q. If the decay function G associated to the family Ktt∈I has a finite 2ν + 1 moment,


then for any locally normal state ρ and each choice of x, y ∈ Λ, the mapping ΨΛ defined in (5.76),satisfies the estimate

(5.85)∑

Z⊂Λ:x,y∈Z

‖ΨΛ(Z, t)‖ ≤ C1(t)F (d(x, y)/3) + C2(t)∑

n=⌊d(x,y)/3⌋

(1 + n)ν+1G(n)

where the time-dependent pre-factors C1 and C2 may be taken as

(5.86) C1(t) = B(t)‖Φp‖F (t) + 4κ2C(t)‖Φq‖F (t)∞∑

n=0

(1 + n)2ν+1G(n),

and C2(t) = 4κ‖F‖C(t)‖Φq‖F (t).It is clear from the statement, as well as the proof, that the estimate proven in Theorem 5.13 does

not require that the mappingsKt satisfy Assumption 5.11. As indicated previously, Assumption 5.11is convenient because it guarantees that the mapping ΨΛ satisfies the continuity requirementsneeded to be a strongly continuous interaction, as defined in the beginning of this subsection.

Proof. Fix Z ⊂ Λ and t ∈ I. A simple norm estimate, using (5.76), shows that

‖ΨΛ(Z, t)‖ ≤ ‖ΠΛZ(Kt(Φ(Z, t)))‖ +

∑

n≥1

∑

X⊂Z:X(n)∩Λ=Z

‖∆ΛX(n)(Kt(Φ(X, t)))‖

≤ B(t)|Z|p‖Φ(Z, t)‖ + 4C(t)∑

n≥1

G(n− 1)∑

X⊂Z:X(n)∩Λ=Z

|X|q‖Φ(X, t)‖.(5.87)

Here we first used that ∆ΛZ(0) = ΠΛ

Z and that ‖ΠΛZ‖ ≤ 1, see Section 4.2 for more details. Next,

we used the local bound on Kt, i.e. (5.78), for the first term on the right-hand-side above. For theremaining terms, we combined the quasi-local bound on Kt, i.e. (5.79), with the estimate (5.8) asproven in Lemma 5.1.

We conclude that∑

Z⊂Λ:x,y∈Z

‖ΨΛ(Z, t)‖ ≤ B(t)∑

Z⊂Λ:x,y∈Z

|Z|p‖Φ(Z, t)‖

+4C(t)∑

n≥0

G(n)∑

X⊂Λ:x,y∈X(n+1)

|X|q‖Φ(X, t)‖.(5.88)

An application of Lemma 8.9 completes the proof.

We finish this subsection with a quasi-locality estimate for the finite-volume dynamics (5.75).Such a result is an immediate consequence of Theorem 3.1 once we obtain that ΨΛ ∈ BF (I) for

some F -function F on (Γ, d). In concrete applications, the existence of such a function F willdepend on the original F -function F and quasi-local decay function G. Rather than make furtherassumptions on F and G from, e.g., Theorem 5.13 let us assume there is an F -function F on (Γ, d)for which

(5.89) max

F (r/3),

∞∑

n=⌊r/3⌋

(1 + n)ν+1G(n)

≤ F (r) for all r ≥ 0 .

We note that in many applications the initial decay functions are weighted F -functions in thesense of Section 8.2, and therefore explicit choices for F are readily determined by manipulatingthe weights. In any case, given such a function F , the bound in (5.85) above implies an explicitpointwise estimate on ‖ΨΛ‖F , see e.g. (5.91) below.

We end this subsection with the following corollary.


Corollary 5.14. Under the assumptions of Theorem 5.13, suppose further that the family Ktt∈Isatisfies Assumption 5.11, and that F is an F -function on (Γ, d) satisfying (5.89). Then, ΨΛ ∈BF (I), and the finite-volume dynamics in (5.75) associated to ΨΛ satisfies the following bound:given any A ∈ AX, B ∈ AY where X,Y ⊂ Λ with X ∩ Y = ∅,

(5.90) ‖[τΛt,s(A), B]‖ ≤ 2‖A‖‖B‖CF

(

e2It,s(ΨΛ) − 1)

∑

x∈X

∑

y∈Y

F (d(x, y))

holds for all s, t ∈ I. Moreover, for s ≤ t,

It,s(ΨΛ) ≤ CF

∫ t

sB(r)‖Φp‖F (r) dr

+4κCF

(

κ

∞∑

n=0

(1 + n)2ν+1G(n) + ‖F‖)

∫ t

sC(r)‖Φq‖F (r) dr(5.91)

where It,s(ΨΛ) is as in (3.21) with F replaced by F .

5.4.3. Results in infinite volume. In this section, we show how the results of the previous sectionextend to the thermodynamic limit. We begin with an assumption on a collection of quasi-localmaps Kt : Aloc

Γ → AΓ. In essence, this definition combines the notion of uniformly locally normalfrom Definition 4.9 with Assumptions 5.11 and 5.12. As always, we consider a quantum latticesystem comprised of (Γ, d) and AΓ, and let I ⊂ R be an interval.

Assumption 5.15. We assume that the family of linear maps Kt : AlocΓ → AΓt∈I , is strongly

continuous, uniformly locally normal, and uniformly quasi-local in the following sense; there isan increasing, exhaustive sequence Λnn≥1 of finite subsets of Γ with a family of linear maps

K(n)t : AΛn → AΛnt∈I for each n ≥ 1 such that:

(i) For each n ≥ 1, the family K(n)t : AΛn → AΛnt∈I satisfies Assumption 5.11.

(ii) There is some p ≥ 0 and a measurable, locally bounded function B : I → [0,∞) for whichgiven any X ∈ P0(Γ) and n ≥ 1 large enough so that X ⊂ Λn,

(5.92) ‖K(n)t (A)‖ ≤ B(t)|X|p‖A‖ for all A ∈ AX and t ∈ I .

(iii) There is some q ≥ 0, a non-negative, non-increasing function G with G(x) → 0 as x → ∞,and a measurable, locally bounded function C : I → [0,∞) for which given any sets X,Y ∈P0(Γ) and n ≥ 1 large enough so that X ∪ Y ⊂ Λn,

(5.93) ‖[K(n)t (A), B]‖ ≤ C(t)|X|q‖A‖‖B‖G(d(X,Y )) for all A ∈ AX , B ∈ AY , and t ∈ I .

(iv) There is some r ≥ 0, a non-negative, non-increasing function H with H(x) → 0 as x→ ∞,and a measurable, locally bounded function D : I → [0,∞) for which given any X ∈ P0(Γ)there exists N ≥ 1 such that for n ≥ N ,

(5.94) ‖K(n)t (A)−Kt(A)‖ ≤ D(t)|X|r‖A‖H(d(X,Γ \ Λn)) for all A ∈ AX and t ∈ I .

Before proving the theorem, we make the following comments. First, if Ktt∈I is a family oflinear maps which satisfies Assumption 5.15, then for any compact I0 ⊂ I, the family Ktt∈I0 isclearly a strongly continuous family of uniformly locally normal maps in the sense of Definition 4.9.Moreover, conditions (ii) and (iii) of Assumption 5.15 guarantee that the sequence of finite-volume

approximates K(n)t n≥1 satisfies Assumption 5.12 with estimates that are uniform in n. In Sec-

tion 6, an explicit family of weighted integral operators of the type discussed in Example 4.11 willbe shown to satisfy all conditions of Assumption 5.15.

Let us now return to the discussion of transformed interactions. Let I ⊂ R be an interval, Φ :P0(Γ)× I → Aloc

Γ be a strongly continuous interaction, and Ktt∈I be a family of transformationssatisfying Assumption 5.15. Let Λnn≥1 be the increasing, exhaustive sequence of finite subsets


of Γ whose existence is guaranteed by Assumption 5.15. For each n ≥ 1, we will denote by HΦΛn

(t)the finite-volume, time-dependent Hamiltonian associated to Φ defined as in (5.72). Let us furtherdenote by

(5.95) K(n)t (HΦ

Λn(t)) =

∑

X⊂Λn

K(n)t (Φ(X, t))

the corresponding finite-volume transformed Hamiltonian. Our assumptions, specifically Assump-tion 5.15 (i), guarantee that the transformed Hamiltonian in (5.95) is still a Hamiltonian in the

sense that t 7→ K(n)t (HΦ

Λn(t)) is strongly continuous and pointwise self-adjoint. As a result, a

finite-volume dynamics may be defined by solving

(5.96)d

dtUn(t, s) = −iK(n)

t (HΦΛn

(t))Un(t, s) with Un(s, s) = 1

and then using the corresponding unitary propagator to declare that

(5.97) τ(n)t,s (A) = Un(t, s)

∗AUn(t, s) for all A ∈ AΛn and t, s ∈ I ,

is the finite-volume time evolution.A main goal of this section is to show that, under appropriate decay assumptions, the finite

volume dynamics in (5.97) converge to a limiting dynamics as n → ∞. To be more precise, let usintroduce some further notation. Fix a locally normal product state ρ on AΓ. As in (5.76), withrespect to this fixed ρ, for any n ≥ 1, Z ⊂ Λn and t ∈ I, set

(5.98) Ψn(Z, t) =∑

m≥0

∑

X⊂Z:X(m)∩Λn=Z

∆Λn

X(m)(K(n)t (Φ(X, t))) .

These finite volume maps Ψn are constructed in such a way that

(5.99) K(n)t (HΦ

Λn(t)) =

∑

X⊂Λn

K(n)t (Φ(X, t)) =

∑

Z⊂Λn

Ψn(Z, t) = HΨn

Λn(t)

and moreover, as checked in Section 5.4.2, under the assumptions above, each Ψn is a stronglycontinuous interaction in the sense of Section 3.1.1. With respect to the same locally normal stateρ, we can also define a map Ψ : P0(Γ)× I → Aloc

Γ by setting

(5.100) Ψ(Z, t) =∑

m≥0

∑

X⊂Z:X(m)=Z

∆X(m)(Kt(Φ(X, t))).

Since the family of transformations Kt locally satisfies Definition 4.9, it is clear that Lemma 4.12applies, and hence Ψ is a strongly continuous interaction as well.

In the remainder of this section, we will show that if the initial interaction Φ decays sufficientlyfast, then the transformed interactions Ψnn≥1 converge locally in F -norm to Ψ in the sense ofDefinition 3.7. Moreover, our assumptions will allow for an application of Theorem 3.8 from whichwe will conclude that the finite-volume dynamics in (5.97) converge. For ease of later reference, letus declare the relevant decay of Φ as an assumption.

Assumption 5.16. Given a ν-regular metric space (Γ, d), and a family of maps Kt : AΓ → AΓt∈Isatisfying Assumption 5.15, we assume Φ is a strongly continuous interaction such that Φm ∈ BF (I)for m = maxp, q, r where p, q, and r are the numbers in Assumption 5.15.

We can now state the main result of this section, for which it will be useful to review Definition 3.7.

Theorem 5.17. Consider a quantum lattice system comprised of a ν-regular metric space (Γ, d)and quasi-local algebra AΓ. Let I ⊂ R be an interval, and F be an F -function on (Γ, d). Assumethat Ktt∈I is a family of linear maps satisfying Assumption 5.15, Φ is an interaction satisfyingAssumption 5.16, and ρ is a locally normal product state on AΓ.


(i) Suppose the quasi-local decay function G from (5.93) has a finite 2ν + 1 moment and F isan F -function on (Γ, d) satisfying (5.89), then Ψ ∈ BF (I).

(ii) Suppose there is some 0 < α < 1 for which Gα has a finite 2ν + 1 moment, where G is as

in (5.93). Suppose also that F is an F -function on (Γ, d) satisfying (5.89) with G replaced

by Gα. Then Ψ ∈ BF (I) and Ψn converges locally in F -norm to Ψ with respect to F .

Some comments are in order. First, under the assumptions of Theorem 5.17(i), the finite-volumeinteractions Ψn, as defined in (5.98), satisfy the assumptions of Theorem 5.13 and hence the estimate

(5.85). In this case, for any F -function F on (Γ, d) satisfying (5.89), the corresponding finite-volume

dynamics, i.e. the automorphisms τ(n)t,s defined in (5.97), satisfy the quasi-locality bound proven

in Corollary 5.14, see (5.90). A main point of Theorem 5.17(i) is that both of these observationsextend to the thermodynamic limit. In fact, the assumptions of Theorem 5.17(i) also guaranteethat the arguments in Theorem 5.13, and hence an analogue of the bound (5.85), also apply tothe infinite-volume interaction Ψ as defined in (5.100). Here we are using that the uniform localconvergence in (5.94) guarantees that both the local bound, see (5.92), and the quasi-local bound,see (5.93), extend to the limiting map Kt, and in this case, Lemma 5.1 applies. Given this, for

any F -function F on (Γ, d) satisfying (5.89), one concludes that Ψ ∈ BF (I). As a result, we canapply Theorem 3.5, where we take the case of trivial on-sites Hz = 0 for all z ∈ Γ. This thenshows that there exists an infinite volume dynamics, which we denote by τt,s, associated to Ψ. Byconstruction, this infinite-volume dynamics τt,s also satisfies Corollary 5.14.

Theorem 5.17(ii) implies that, under the slightly stronger decay assumptions, the finite-volume

dynamics τ(n)t,s converge to the infinite-volume dynamics τt,s in the sense given by Theorem 3.8.

Since the interactions Ψn are constructed using finite-volume local decompositions, see (5.98), theyare not finite-volume restrictions of Ψ, and so an additional argument is required here. We remarkthat the decay assumptions in Theorem 5.17(ii) imply the decay assumed in Theorem 5.17(i). Asa result, the better quasi-locality estimates for the dynamics, which follow as a consequence of theassumptions in Theorem 5.17(i), may be used generally.

Next, a careful look at the proof of Theorem 5.17(i) below shows that we actually only requireΦm ∈ BF (I) for m = max(p, q). The proof of Theorem 5.17(ii) requires the stronger condition ofAssumption 5.16, namely Φm ∈ BF (I) for m = max(p, q, r).

Finally, we note that if the decay function G in (5.93) is a weighted F -function, the argumentsbelow can be simplified a bit.

Proof. The proof of Theorem 5.17(i) is argued in the paragraphs above.To prove Theorem 5.17(ii), first note that as 0 < α < 1, it is clear that finiteness of the 2ν + 1

moment of Gα implies finiteness of the 2ν + 1 moment of G. In this case, the estimate proven inTheorem 5.13, see (5.85), holds for each finite-volume interaction Ψn as well as for Ψ. Since G isnon-negative and non-increasing, G(n) ≤ G1−α(0)Gα(n), and therefore, given any [a, b] ⊂ I,

supn≥1

∫ b

a‖Ψn‖F (r) dr ≤

∫ b

aB(r)‖Φp‖F (r) dr

+4κ

(

κ

∞∑

n=0

(1 + n)2ν+1G(n) + ‖F‖G1−α(0)

)

∫ b

aC(r)‖Φq‖F (r) dr(5.101)

holds for any F -function F satisfying the conditions of Theorem 5.17(ii). An analogous boundholds for the infinite volume interaction Ψ.

We need only show that Ψn converges locally in F -norm to Ψ with respect to F , see Definition 3.7.Let Λ ∈ P0(Γ) and take n ≥ 1 large enough so that Λ ⊂ Λn. For any Z ⊂ Λ and each t ∈ I, weestimate

(5.102) ‖Ψn(Z, t) −Ψ(Z, t)‖ ≤ Σ1(Z, t) + Σ2(Z, t) + Σ3(Z, t)


where, for the terms corresponding to m = 0 in (5.98) and (5.100), we have set

(5.103) Σ1(Z, t) = ‖ΠΛn

Z (K(n)t (Φ(Z, t))) −ΠZ(Kt(Φ(Z, t)))‖ ,

we have collected the bulk of the terms in

(5.104) Σ2(Z, t) =∑

m≥1

∑

X⊂Z:X(m)=Z

‖∆Λn

X(m)(K(n)t (Φ(X, t))) −∆X(m)(Kt(Φ(X, t)))‖ ,

and finally, we have denoted any boundary terms by

(5.105) Σ3(Z, t) =∑

m≥1

∑

X⊂Z:X(m)6⊆Z,X(m)∩Λn=Z

‖∆Λn

X(m)(K(n)t (Φ(X, t)))‖ .

It is now clear that

(5.106)∑

Z⊂Λ0x,y∈Z

‖Ψn(Z, t)−Ψ(Z, t)‖ ≤3∑

j=1

∑

Z⊂Λ0x,y∈Z

Σj(Z, t).

To complete this proof, we will show that each of the 3 sums above are bounded by a product of:a) a measurable, locally bounded function of t; b) F (d(x, y)); and c) a quantity that vanishes asn→ ∞. Given this, it is clear that Ψn converges to Ψ locally in F -norm.

Consider the first collection of terms. By consistency of the projections,

(5.107) Σ1(Z, t) ≤ ‖K(n)t (Φ(Z, t)) −Kt(Φ(Z, t))‖ ≤ D(t)|Z|r‖Φ(Z, t)‖H(d(Z,Γ \ Λn)).

Here, we have also applied Assumption 5.15(iv). Since H is non-increasing, the bound

(5.108)∑

Z⊂Λ0x,y∈Z

Σ1(Z, t) ≤ D(t)‖Φr‖F (t)F (d(x, y))H(d(Λ0 ,Γ \ Λn))

follows as Φr ∈ BF (I). Since F maximizes F , this completes the argument for the first set of terms.We now consider the bulk terms. An application of Corollary 5.3, see (5.17), yields

‖∆Λn

X(m)(K(n)t (Φ(X, t))) −∆X(m)(Kt(Φ(X, t)))‖

≤ min

2‖K(n)t (Φ(X, t)) −Kt(Φ(X, t))‖, 8C(t)|X|qG(m− 1)

≤ 2‖Φ(X, t)‖min D(t)|X|rH(d(X,Γ \ Λn)), 4C(t)|X|qG(m− 1) .(5.109)

To obtain an estimate with explicit decay in both n and m, we use the naive bound mina, b ≤a1−αbα which is valid for any 0 < α < 1 and all non-negative a and b. If we denote by dn =d(Λ0,Γ \ Λn) and p

′ = max(q, r), then the right-hand-side of (5.109) may be further estimated by

(5.110) 22α+1|X|p′‖Φ(X, t)‖D(t)1−αH(dn)1−αC(t)αG(m− 1)α .

Using this, we conclude that

(5.111)∑

Z⊂Λ0x,y∈Z

Σ2(Z, t) ≤ 22α+1D(t)1−αC(t)α(t)H(dn)1−α

∑

m≥0

Gα(m)∑

X⊂Λ0:

x,y∈X(m+1)

|X|p′‖Φ(X, t)‖,

and Lemma 8.9 applies. Recalling how F is defined, this completes the argument for the secondcollection of terms.


For the final collection of terms, each non-zero contribution must correspond to values of m ≥ 1large enough so that X(m) ∩ (Γ \ Λn) 6= ∅. As such, using the notation above, one checks thatm ≥ dn = d(Λ0,Γ \ Λn). The bound (5.8) applies to each term and thus

(5.112)∑

Z⊂Λ0x,y∈Z

Σ3(Z, t) ≤ 4C(t)∑

m≥dn

G(m− 1)∑

X⊂Λ0:

x,y∈X(m)

|X|q‖Φ(X, t)‖.

Exploiting again that G = G1−αGα and using its non-increasing behavior, we obtain decay in n.Estimating what remains using Lemma 8.9, we have completed the proof of Theorem 5.17(ii).

5.5. Quasi-locality for the difference of two dynamics. In this section, we prove a quasi-locality estimate for the difference of two dynamics as discussed in Example 5.6 of Section 5.2.

Theorem 5.18. Let (Γ, d) be a ν-regular metric space. Fix a collection of densely defined, self-adjoint on-site Hamiltonians Hzz∈Γ and two time-dependent interactions Φ,Ψ ∈ BF (I). For anyΛ ∈ P0(Γ) and each t ∈ I, consider the Hamiltonians

(5.113) H(Φ)Λ (t) =

∑

z∈Λ

Hz +∑

Z⊂Λ


∑

z∈Λ

Hz +∑

Z⊂Λ

Ψ(Z, t).

For any s, t ∈ I, denote by τΛt,s and αΛt,s the dynamics corresponding to H

(Φ)Λ and H

(Ψ)Λ , respectively,

and define KΛt,s : AΛ → AΛ by

(5.114) KΛt,s(A) = τΛt,s(A)− αΛ

t,s(A).

If F has a finite 2ν-moment, i.e.∑∞

n=0(1 + n)2νF (n) <∞, then for any X,Y ⊂ Λ

(5.115) ‖[KΛt,s(A), B]‖ ≤ 4C−1

F It,s(Φ−Ψ)‖A‖‖B‖minC(1)t,s ‖F‖|X|, C(2)

t,s G(d(X,Y ))for any A ∈ AX , B ∈ AY , and t, s ∈ I. Here one may take

(5.116) C(1)t,s = e2min(It,s(Φ),It,s(Ψ)) and C

(2)t,s =

(

C(1)t,s − 1

)

(

1 +5‖F‖CF

)

+ κ2

and with R = d(X,Y ) we find that

G(R) = (1 + |X(R/2)|)GF (X,Λ \X(R/2)) + |X(R/2)|GF (X(3R/8), Y )

+|X(R/2)|∞∑

n=⌊R/4⌋

(1 + n)2νF (n).(5.117)

Proof. To begin, we note that for any X,Y ∈ P0(Γ), the naive commutator bound

(5.118) ‖[KΛt,s(A), B]‖ ≤ 2‖KΛ

t,s(A)‖‖B‖holds for any A ∈ AX , B ∈ AY , and s, t ∈ I. In this case, the local bound proven in Theorem 3.4(i),see also (5.33), provides a rough estimate, which is linear in It,s(Φ−Ψ). This explains the first partof the minimum in (5.115). Given this, we need only consider the case of d(X,Y ) > 0. Moreover,as is clear from the arguments given in the proof of Theorem 3.3, we need only consider the caseof trivial on-sites, i.e. Hz = 0 for all z ∈ Γ.

Let X,Y ∈ P0(Γ) satisfy d(X,Y ) > 0 and, for convenience, assume that s ≤ t. Writing KΛt,s(A)

as in (3.70), the bound

(5.119) ‖[KΛt,s(A), B]‖ ≤

∑

Z⊂Λ

∫ t

s‖[τΛr,s([αΛ

t,r(A),Θ(Z, r)]), B]‖ dr

follows readily; see also (3.72). Here, as in (3.71), we have denoted by Θ the time-dependentinteraction with terms Θ(Z, r) = Φ(Z, r)−Ψ(Z, r).


In the estimates below, we use an argument similar to that of Theorem 3.4, see in particular

(3.74), to show that the claim holds with e2It,s(Ψ) replacing C(1)t,s in the definition of C

(2)t,s . However,

by reordering the dynamics in (5.114), (or equivalently, by considering −KΛt,s(A)) we see that the

analogue of (5.119) holds with the roles of the dynamics τt,s and αt,s interchanged. Since the

argument given below applies equally well in this case, it will be clear that C(2)t,s can then be

expressed in terms of C(1)t,s . We now continue with our estimate of the right-hand-side of (5.119).

To prove (5.115), we first consider those terms on the right-hand-side of (5.119) correspondingto Z ⊂ Λ with d(Z,X) > d(X,Y )/2. Since τΛr,s is an automorphism, the commutator bound

(5.120) ‖[τΛr,s([αΛt,r(A),Θ(Z, r)]), B]‖ ≤ 2‖B‖‖[αΛ

t,r(A),Θ(Z, r)]‖

is clear. By Theorem 3.3, the dynamics αΛt,r corresponding to H

(Ψ)Λ satisfies a quasi-locality bound,

in particular, we may estimate as in (5.28). In this case, an application of Corollary 8.5 withR = d(X,Y )/2 shows that

∑

Z⊂Λ:

d(Z,X)>R

∫ t

s‖[τΛr,s([αΛ

t,r(A),Θ(Z, r)]), B]‖ dr ≤

4‖A‖‖B‖CF

(e2It,s(Ψ) − 1)It,s(Θ)GF (X,Λ \X(R)).(5.121)

We need only estimate those terms on the right-hand-side of (5.119) corresponding to Z ⊂ Λwith d(Z,X) ≤ d(X,Y )/2. For these terms, we first make a strictly local approximation of theinner-most dynamics, i.e. αΛ

t,r. Given the quasi-locality estimate (5.28) for αΛt,r, an application of

Lemma 5.1 shows that

(5.122) ‖αΛt,r(A)−AR(r)‖ ≤ 4‖A‖

CF(e2It,s(Ψ) − 1)GF (X,Λ \X(R))

where we have set AR(r) = ΠΛX(R)(α

Λt,r(A)) and found an upper bound independent of s ≤ r ≤ t.

In this case, for any s ≤ r ≤ t,

‖[τΛr,s([αΛt,r(A),Θ(Z, r)]), B]‖ ≤ ‖[τΛr,s([AR(r),Θ(Z, r)]), B]‖ +

+‖[τΛr,s([αΛt,r(A) −AR(r),Θ(Z, r)]), B]‖.(5.123)

For the second term on the right-hand-side of (5.123), it is clear that

(5.124) ‖[τΛr,s([αΛt,r(A)−AR(r),Θ(Z, r)]), B]‖ ≤ 16‖A‖‖B‖

CF‖Θ(Z, r)‖(e2It,s(Ψ)−1)GF (X,Λ\X(R))

and therefore, the bound

∑

Z⊂Λ:

d(Z,X)≤R

∫ t

s‖[τΛr,s([αΛ

t,·(A)−AR(r),Θ(Z, r)]), B]‖ dr ≤

16‖A‖‖B‖‖F‖C2F

(e2It,s(Ψ) − 1)It,s(Θ)|X(R)|GF (X,Λ \X(R))(5.125)

follows from an application of Proposition 8.4, see (8.41).With the remaining terms, i.e. those corresponding to the first term on the right-hand-side of

(5.123), we find it useful to further sub-divide the sets Z into those of relative ‘large’ and ‘small’diameter. More precisely, we will estimate using

(5.126)∑

Z⊂Λ:

d(Z,X)≤R

· ≤∑

x∈X(R)

∑

Z⊂Λ:x∈Z

diam(Z)≤R/2

·+∑

x∈X(R)

∑

Z⊂Λ:x∈Z

diam(Z)>R/2

·


For the terms with ‘small’ diameter, we apply the quasi-locality estimate for the outer dynamicsτΛt,s, again we use the form found in (5.28), to obtain

(5.127) ‖[τΛr,s([AR(r),Θ(Z, r)]), B]‖ ≤ 4‖A‖‖B‖CF

‖Θ(Z, r)‖(e2It,s(Φ) − 1)GF (X(R) ∪ Z, Y ).

Clearly, X(R) ∪ Z ⊂ X(3R/2) for any Z with Z ∩X(R) 6= ∅ and diam(Z) ≤ R/2. In this case,

∑

x∈X(R)

∑

Z⊂Λ:x∈Z

diam(Z)≤R/2

∫ t

s‖[τΛr,s([AR(r),Θ(Z, r)]), B]‖ dr ≤

4‖A‖‖B‖‖F‖C2F

(e2It,s(Φ) − 1)It,s(Θ)|X(R)|GF (X(3R/2), Y )(5.128)

follows immediately from the arguments in Proposition 8.6, see (8.55).The remaining terms have relatively large diameter, and so we make the naive estimate

(5.129) ‖[τΛr,s([AR(r),Θ(Z, r)]), B]‖ ≤ 4‖A‖‖B‖‖Θ(Z, r)‖.As a consequence,

∑

x∈X(R)

∑

Z⊂Λ:x∈Z

diam(Z)>R/2

∫ t

s‖[τΛr,s([AR(r),Θ(Z, r)]), B]‖ dr ≤

4κ2‖A‖‖B‖CF

It,s(Θ)|X(R)|[M2ν (F )](R/2)(5.130)

follows from Proposition 8.6, see (8.56).Collecting the estimates in (5.121), (5.125), (5.128), and (5.130), we find (5.115) as claimed.

6. The spectral flow

In this section, we consider a family of finite volume quantum lattice Hamiltonians HΛ(s) actingon a Hilbert spaceHΛ that depend smoothly on a parameter s ∈ [0, 1]. We assume that the spectrumof HΛ(s) can be decomposed into two non-empty disjoint sets, i.e. spec(HΛ(s)) = Σ1(s) ∪ Σ2(s),where Σ1(s) is bounded, and the distance between Σ1(s) and Σ2(s) is greater than a positive valueindependent of s. The main goal of this section is to show that if the interaction defining HΛ(s)is smooth and decays sufficiently fasts, then we can use the theory of Section 5 to construct aquasi-local automorphism αs : AΛ → AΛ, which we call the spectral flow, that maps the spectralprojection of HΛ(s) onto Σ1(s) back to the spectral projection of HΛ(0) onto Σ1(0). In Section 7we use the spectral flow to discuss the classification of gapped ground state phases. A secondimportant application concerns models with a spectral gap above their ground states, for which sparameterizes a perturbation of the system; this is the main topic we analyze in [96]. While boththese applications are for ground states, the methods we introduce here are more general and workequally well for isolated bounded subsets anywhere in the spectrum.

Denoting by P (s) the spectral projection of HΛ(s) onto Σ1(s), the existence of an automorphismαs satisfying

(6.1) αs(P (s)) = P (0)

is well-known. As shown by Kato in [67], under certain conditions which guarantee the smoothnessof P (s), the unique strong solution of

(6.2)d

dsUK(s) = −i[P ′(s), P (s)]UK(s), UK(0) = 1

is unitary and satisfiesαKs (P (s)) := UK(s)∗P (s)UK(s) = P (0).


The automorphism studied by Kato was for a family of Hamiltonians defined on a general Hilbertspace H, and so his results do not take into account the locality structure of a quantum latticesystem. As a result, the automorphism induced by UK(s) is not obviously quasi-local. Hastings andWen were the first to introduce a technique for constructing an automorphism on a quantum latticesystem that both satisfies (6.1) and is quasi-local [59]. In that work, they referred to the quasi-local automorphism as the quasi-adiabatic evolution (or continuation). It is this approach thatwe follow in this section. Neither name, spectral flow or quasi-adiabatic continuation, accuratelyand unambiguously captures the essence of this quasi-local automorphism. It suffices to say thatit is a unitary dynamics with useful properties. In other works, Hastings introduced novel ways tocombine particular instances of the spectral flow with quasi-locality properties of quantum latticesystems, most notably in [54]. This work inspired a string of new results in the theory of quantumlattice systems, and so it seems appropriate to refer to the generator of this spectral flow as theHastings generator.

6.1. Set up and main results. We first consider a family of parameter dependent Hamiltonianson a general complex Hilbert space H and later return to apply our results to the setting of quantumlattice systems. Specifically, we consider operators that depend on a parameter s ∈ [0, 1], and wenote that the choice of interval [0, 1] is a matter of convenience. We begin with the followingdefinition.

Definition 6.1. Let H be a complex Hilbert space. We say that a map Φ : [0, 1] → B(H) isstrongly C1 if Φ(s) is strongly differentiable for all s ∈ [0, 1], and the derivative Φ′ : [0, 1] → B(H)is continuous in the strong operator topology.

We consider a family of parameter dependent Hamiltonians of the form

(6.3) H(s) = H +Φ(s), s ∈ [0, 1]

where H is a self-adjoint operator acting on some dense domain D ⊂ H, and Φ : [0, 1] → B(H) isstrongly C1 and pointwise self-adjoint, i.e. Φ(s)∗ = Φ(s) for all 0 ≤ s ≤ 1. Since Φ is boundedand self-adjoint, for each s ∈ [0, 1] it is clear that H(s) corresponds to a well-defined, self-adjointoperator with the same dense domain D ⊂ H. We will refer to H(s)s∈[0,1] as a smooth family ofHamiltonians on H.

For each 0 ≤ s ≤ 1, let us denote by τ(s)t the Heisenberg dynamics corresponding to H(s), i.e.,

(6.4) τ(s)t (A) = eitH(s)Ae−itH(s) for all A ∈ B(H) and t ∈ R .

It is clear that for each s, this dynamics is a one-parameter family of automorphisms of B(H), andso for any real-valued function W ∈ L1(R), the mapping D : [0, 1] → B(H) given by

(6.5) D(s) =

∫

R

τ(s)t (Φ′(s))W (t) dt

is well-defined, pointwise self-adjoint, bounded, and continuous in the strong operator topology. Inthis case, the methods of Section 2.2 show that the unique strong solution of

(6.6)d

dsU(s) = −iD(s)U(s) with U(0) = 1l

is well-defined, unitary, and norm-continuous. In terms of these unitaries, we can define an auto-morphism αs : B(H) → B(H) for each 0 ≤ s ≤ 1 by

(6.7) αs(A) = U(s)∗AU(s).

Note that here, D(s), U(s), and αs all depend on the choice of weight function W ∈ L1(R). Wewill use this construction to define the spectral flow of interest.

As described in the introduction, we will consider the situation that the smooth family of Hamil-tonians defined as in (6.3) has a spectrum which can be decomposed into two disjoint, non-empty


sets. This decomposition will depend on 0 ≤ s ≤ 1, and we are particularly interested in caseswhere the gap between these sets has a uniform lower bound. To be precise, some additional no-tation will be convenient: for any two non-empty sets X,Y ⊂ R, denote by d(X,Y ) the distancebetween these sets:

(6.8) d(X,Y ) := inf|x− y| : x ∈ X and y ∈ Y .Assumption 6.2. For each 0 ≤ s ≤ 1, the spectrum of H(s) can be partitioned into two disjointsets Σ1(s) and Σ2(s), i.e. spec(H(s)) = Σ1(s) ∪ Σ2(s), such that

(6.9) γ′ := inf0≤s≤1

d(Σ1(s),Σ2(s)) > 0,

and, moreover, there are compact intervals I(s) with end-points depending smoothly on s, for whichΣ1(s) ⊂ I(s) ⊂ (R \ Σ2(s)) and µ(s) := d(I(s),Σ2(s)) satisfies µ := inf0≤s≤1 µ(s) > 0.

In many concrete examples one can pick the interval I(s) as the smallest interval containingΣ1(s), and in that case µ = γ′.

Given a smooth family of Hamiltonians H(s) that satisfy Assumption 6.2, the spectral flow ofinterest depends on the choice of an auxiliary parameter 0 < γ ≤ γ′. For any such γ and 0 ≤ s ≤ 1,we define the spectral flow αγ

s : B(H) → B(H) by

(6.10) αγs (A) = Uγ(s)∗AUγ(s)

where Uγ(s) is the unitary solution to (6.6) for the self-adjoint operator

(6.11) Dγ(s) =

∫

R

τ(s)t (Φ′(s))Wγ(t) dt

defined by a well-chosen γ-dependent, real-valued weight function Wγ ∈ L1(R). In Section 6.2 westate the necessary conditions for choosing Wγ and give an explicit example a weight function thatsatisfies these conditions. In fact, we will be able to define weight functions Wγ for any γ > 0.However, to obtain the spectral flow property, i.e. (6.1), one must choose Wγ with 0 < γ ≤ γ′. Asdiscussed in the introduction, the Hamiltonian Dγ(s) will be called a Hastings generator. We cannow state the first main result of this section.

Theorem 6.3. Let H be a complex Hilbert space, and H(s) be a smooth family of Hamiltonians asin (6.3) satisfying Assumption 6.2. For any 0 < γ < γ′, there is a real-valued function Wγ ∈ L1(R)such that the automorphism αγ

s : B(H) → B(H) defined as in (6.10) satisfies

(6.12) αγs (P (s)) = P (0)

for all 0 ≤ s ≤ 1. Here, P (s) denotes the spectral projection associated to H(s) onto the isolatedpart of the spectrum Σ1(s).

In the context of a quantum lattice system, the novel feature of the Hastings generator is thatit generates a quasi-local family of automorphisms. This is the second main result of this section.Recall that given a quantum lattice system (Γ, d) and AΓ, the local Hamiltonians for a stronglycontinuous interaction Φ : P0(Γ)× [0, 1] → Aloc

Γ are given by

(6.13) HΛ(s) =∑

X⊆Λ

Φ(X, s), for all Λ ∈ P0(Γ).

Note that if Φ(X, s) is strongly C1 for all X ⊂ Λ in the sense of Definition 6.1, then the HamiltonianHΛ(s) is also strongly C

1. In this case, for every Λ ∈ P0(Γ) we may define the finite volume Hastingsgenerator by

(6.14) DγΛ(s) =

∫

R

τ(s)t (H ′

Λ(s))Wγ(t) dt,


where for each s ∈ [0, 1], τ(s)t is the Heisenberg dynamics associated to HΛ(s). We may now state

the quasi-locality result.

Theorem 6.4. Consider a quantum lattice system comprised of ν-regular metric space (Γ, d) andquasi-local algebra AΓ. Suppose that Φ ∈ BF ([0, 1]) for an F -function of the form

(6.15) F (r) = e−arθ (1 + r)−p for some a > 0, 0 < θ ≤ 1 and p > ν + 1.

If Φ(X, s) is strongly C1 for all X ∈ P0(Γ) and Φ′1 ∈ BF ([0, 1]) where Φ′

1(X, s) = |X|Φ′(X, s), then

for any γ > 0 there is an F -function, F (γ), such that for any Λ ∈ P0(Γ)

(6.16) DγΛ(s) =

∑

X⊆Λ

ΨΛ(X, s)

for a strongly-continuous interaction ΨΛ ∈ BF (γ)([0, 1]). Moreover, there is an interaction Ψ ∈BF (γ)([0, 1]) such that ΨΛn converges locally in F -norm to Ψ with respect to F (γ) for any sequenceof increasing and absorbing finite volumes Λn ↑ Γ.

We give some context for this result. Suppose that Φ ∈ BF ([0, 1]) is such that the local Hamil-tonians HΛ(s) are strongly C1. Recall that for any γ > 0, the Hastings generator Dγ

Λ(s), which isdefined in terms of H ′

Λ(s) (see (6.14)), is strongly continuous and self-adjoint. The automorphism

αγ,Λs defined as in (6.10) can then be recognized as the Heisenberg dynamics associated to DΛ(s).

If Theorem 6.4 holds, then DΛ(s) is itself a local Hamiltonian associated to a strongly-continuous

interaction ΨΛ ∈ BF (γ)([0, 1]). Applying the Lieb-Robinson bound, i.e. Theorem 3.1, to αγ,Λs shows

that the spectral flow is quasi-local as claimed. In the proof of Theorem 6.4, we show that the norm‖ΨΛ‖F (γ) is bounded from above by a constant independent of Λ, from which local F -norm con-vergence will follow. The interaction Ψ then defines an infinite volume spectral flow automorphismαγs : AΓ → AΓ that is also quasi-local.Note that we do not require Assumption 6.2 for Theorem 6.4, and in particular, the quasi-

locality result holds where the spectrum of HΛ(s) is or is not gapped. If, however, HΛ(s) satisfies

Assumption 6.2 with gap γ′Λ > 0, then the finite-volume automorphisms αΛ,γs generated by Dγ

Λ(s)for any 0 < γ ≤ γ′Λ will both be quasi-local and satisfy (6.12). In applications to stability, one isinterested in the situation that there is some sequence of finite volumes (Λn)n≥1 for which bothTheorem 6.3 and Theorem 6.4 hold simultaneously and that the gaps γ′Λn

as in (6.9) are uniformlybounded from below by a positive constant independent of n.

In what follows, we will typically work with a Hastings generator and spectral flow automorphismthat depend on a fixed value of γ. As such, we will often suppress the dependence of γ from ournotation.

The remainder of the section is organized as follows. In Section 6.2 we define the explicit weightfunction Wγ used in our results and prove some basic decay estimates on this function. The readercan skip these details on first reading. Recall that the definition of the Hastings generator is givenin terms of a specific weighted integral operator. In Section 6.3 we define several general weightedintegral operators in terms of appropriate L1 functions and prove some useful properties. We usethe results from this section to give the proof of Theorem 6.3 in Section 6.4. We consider quantumlattice systems in Section 6.5 where we show that, in this context, the weighted integral operatorsintroduced in Section 6.3 are quasi-local when defined using the weight functions from Section 6.2.We then use these results to prove Theorem 6.4 (which is restated as Theorem 6.14). We end thesection by showing that there is a well-defined spectral flow automorphism in the thermodynamiclimit when the conditions of Theorem 6.4 hold.

6.2. An explicit weight. To write down the generator of the spectral flow dynamics, see (6.11)above, requires a weight function with certain properties. In Section 6.3 we will define a class of


transformations on the algebra of observables of the form

I(A) =

∫

R

w(t)τt(A) dt

where w ∈ L1(R). In the following section we will make increasingly detailed assumptions of w inorder to prove useful properties of the map I. At some point, it becomes more efficient to workwith a specific family of functions w for which the assumptions hold. Having such a family offunctions will make it possible to state explicit decay estimates that are useful for applications.As such, in this section we introduce this family of functions, for which interesting properties werealready investigated in [60], and prove some basic estimates; these will be particularly relevant inSection 6.5.2. It will be clear that other functions can be used to derive similar results. The detailsof this section can be skipped on first reading. Its main importance is to demonstrate the existenceof functions with all the desired properties.

Consider the sequence (an)n≥1 defined by

(6.17) an =a1

n ln(n)2for n ≥ 2 and

∞∑

n=1

an =1

2.

In terms of this sequence, define a function w : R → R by setting

(6.18) w(0) = c and w(t) = c∞∏

n=1

(

sin(ant)

ant

)2

if t 6= 0

where c > 0 is chosen so that

(6.19)

∫

R

w(t) dt = 1 .

It follows from Lemma 6.5 below that w ∈ L1(R) and so this constant is well-defined.It is clear that w is non-negative and even. Moreover, if we denote by wγ : R → R the unitary

Fourier transform of w, i.e. for each k ∈ R

(6.20) wγ(k) =1√2π

∫

R

e−iktwγ(t) dt ,

then it is easy to check, see e.g [12], that supp(w) ⊂ [−1, 1]. The following lemma provides a usefulestimate on w.

Lemma 6.5. Let a > 0 and p ≥ 0 be an integer. For any x > 1 with ln(x) ≥ max

9,√

p+1a

, one

has that

(6.21)

∫ ∞

x

(

t

ln(t)2

)p

e− at

ln(t)2 dt ≤ 9(p+ 2)

7a

(

x

ln(x)2

)p+1

e− ax

ln(x)2 .

As a consequence, there is a number η ∈ (2/7, 1) for which if x ≥ e9, then

(6.22)

∫ ∞

xw(t) dt ≤ 27

14ce4(

x

ln(x)2

)2

e− ηx

ln(x)2 .

Proof. To see (6.21), consider the change of variables: u = at/ ln(t)2. Clearly,

(6.23)du

dt=

(

1− 2

ln(t)

)

a

ln(t)2.

It will be convenient to take x large enough so that ln(x)4 ≤ x. As one readily checks, this is thecase if x ≥ e9; however, we note that this lower bound is not optimal. In any case, using this one


also has that

(6.24)du

dt≥ 7a2

9uif t ≥ e9 .

Consequently,

(6.25)

∫ ∞

x

(

t

ln(t)2

)p

e− at

ln(t)2 dt ≤ 9

7ap+2

∫ ∞

u(x)up+1e−u du.

For integers p ≥ 0, the above integral may be bounded using

(6.26)

∫ ∞

kup+1e−u du = (p+ 1)!e−k

p+1∑

n=0

kn

n!≤ (p + 2)kp+1e−k

where the final inequality is valid whenever k ≥ p + 1. With the further constraint that ln(x) ≥√

p+1a , the bound

(6.27) p+ 1 ≤ a ln(x)2 ≤ u(x)

follows, using again that ln(x)4 ≤ x. Now (6.21) follows from (6.25) and (6.26).We now estimate w to establish (6.22). Note that for any N ≥ 1 and t 6= 0,

(6.28) w(t) ≤ c

N∏

n=1

(

sin(ant)

ant

)2

≤ c

(a1t)2N

N∏

n=2

n ln(n)2 ≤ c

(

ln(N)2

a1t

)2N

(N !)2.

Using Stirling’s formula, i.e. N ! ≤ eNN+ 12 e−N , and choosing N = ⌊ a1t

ln(t)2⌋, we find that

w(t) ≤ ce2(

N · ln(N)2

a1t

)2N

Ne−2N

≤ ce2

(

1

ln(t)2· ln(

a1t

ln(t)2

)2)2N

a1t

ln(t)2· e−2(

a1t

ln(t)2−1)

≤ ce4a1t

ln(t)2· e−

2a1t

ln(t)2(6.29)

where, for the final inequality above, we used that t is large enough so that both 1 ≤ ln(t)2 andln(a1t)

2 ≤ ln(t)2 hold. Since (6.17) implies that a1 < 1/2, both inequalities are true if t ≥ e. As

(6.30) 1 +

∞∑

n=2

1

n ln(n)2≤ 1 +

1

2 ln(2)2+

∫ ∞

2

1

t ln(t)2dt ≤ 3.5 ,

it is clear that a1 > 1/7. Now, setting η = 2a1, we have found that

(6.31) w(t) ≤ cηe4

2· t

ln(t)2· e−

ηt

ln(t)2 for all t ≥ e .

Now (6.22) follows from (6.21).

For our estimates on the spectral flow, it will be convenient to rescale this weight function w.For any γ > 0, define wγ : R → R by setting

(6.32) wγ(t) = γw(γt) .

It is clear that each such wγ is non-negative, even, L1-normalized, and moreover,

(6.33) supp(wγ) ⊂ [−γ, γ] .


The function Wγ : R → R given by

(6.34) Wγ(x) = −∫ x

−∞wγ(t) dt+H(x) for x ∈ R,

where H(x) is the Heavyside function (for clarity, we take H(0) = 1) will also play a key role below.This may be re-written as

(6.35) Wγ(0) =1

2and Wγ(x) = sgn(x) ·

∫ ∞

|x|wγ(t) dt for x 6= 0.

Thus Wγ is odd, and since wγ is normalized and even, one has that ‖Wγ‖∞ ≤ 1/2. In fact, a shortcalculation shows that

(6.36) ‖Wγ‖1 =∫

R

|Wγ(t)| dt =2

γ

∫ ∞

0tw(t) dt .

It is clear from (6.34) that the distributional derivative of Wγ is

(6.37)d

dxWγ(x) = −wγ(x) + δ0(x)

and thus its (unitary) Fourier transform satisfies

(6.38) Wγ(0) = 0 and (ik)Wγ(k) = −wγ(k) +1√2π

for all k 6= 0.

In particular, we have

(6.39) Wγ(k) =−i√2πk

, for k /∈ (−γ, γ).

As we will see in subsequent sections, a “well-chosen” weight function Wγ for defining the spectralflow as described following (6.11) is one which satisfies (6.39) and has a decay estimate that is atleast stretched exponential, similar to the next result.

Corollary 6.6. Let γ > 0. If γx ≥ e9, then

(6.40)

∫ ∞

xwγ(t) dt ≤

27

14ce4(

γx

ln(γx)2

)2

e−η γx

ln(γx)2

with c as in (6.18), see also (6.19), and η ∈ (2/7, 1) as in Lemma 6.5. Moreover, if γx ≥ e9, then

(6.41)

∫ ∞

xWγ(t) dt ≤

486

49γηce4(

γx

ln(γx)2

)3

e−η γx

ln(γx)2

again with c and η as above.

6.3. On weighted integrals of dynamics. In this section, we briefly discuss some general factsabout weighted integrals of a dynamics. Such operators arise as the generator of the spectral flow,and in this case, a number of their basic properties are relevant.

6.3.1. Some generalities. Let H be a densely defined self-adjoint operator on a Hilbert space H.Denote by τt the associated Heisenberg dynamics, i.e. the one parameter family of automorphismsof B(H) given by

(6.42) τt(A) = eitHAe−itH for any A ∈ B(H) and all t ∈ R .

For any w ∈ L1(R), a bounded mapping I : B(H) → B(H) is defined by setting

(6.43) I(A) =

∫

R

τt(A)w(t) dt for any A ∈ B(H) .


In fact, Stone’s theorem guarantees that this integral is well-defined in both the weak and strongsense. We refer to the operator I above as the integral of the dynamics τt weighted by w, or morebriefly, as a weighted integral operator.

Our applications will mainly concern families of these weighted integral operators. In fact,supposeH(s) = H+Φ(s) is as described in (6.3) and for each 0 ≤ s ≤ 1, consider Is : B(H) → B(H)with

(6.44) Is(A) =

∫

R

τ(s)t (A)w(t) dt

here τ(s)t is the dynamics corresponding to H(s), see (6.3) and (6.4), and w ∈ L1(R) is real-valued.

The following lemma is a useful observation.

Lemma 6.7. Let H be a densely defined self-adjoint operator on a Hilbert space H and, for s ∈[0, 1], let Φ(s) = Φ(s)∗ ∈ B(H) be continuous in s for the strong operator topology. Supposew ∈ L1(R) is real-valued, and A : [0, 1] → B(H) is pointwise self-adjoint and continuous in thestrong operator topology. Then, the mapping D : [0, 1] → B(H) given by

(6.45) D(s) = Is(A(s)) =

∫

R

τ(s)t (A(s))w(t) dt

is pointwise self-adjoint and continuous in the strong operator topology.

Proof. Self-adjointness of D(s), which uses that w is real-valued, is clear. Set A(s, t) ∈ B(H) by

(6.46) A(s, t) = τ(s)t (A(s)) = eitH(s)A(s)e−itH(s) for any 0 ≤ s ≤ 1 and t ∈ R .

With s0 ∈ [0, 1] fixed, for any 0 ≤ s ≤ 1, we have that

(6.47) ‖(D(s)−D(s0))ψ‖ ≤∫

R

‖(A(s, t)−A(s0, t))ψ‖ |w(t)| dt for any ψ ∈ H .

Stone’s theorem guarantees that for each 0 ≤ s ≤ 1, the mapping A(s, ·) : R → B(H) is continuousin the strong operator topology, and so the integrand above is clearly measurable. We now claimthat for each t ∈ R, A(·, t) : [0, 1] → B(H) is also continuous in the strong operator topology. Giventhis, the claimed continuity of D will follow from an application of dominated convergence. Herewe are using that strong continuity of A implies sup0≤s≤1 ‖A(s)‖ <∞.

Due to the form of A(s, t), see (6.46), we need only show that s 7→ eitH(s) is strongly continuousfor each fixed t ∈ R. To see this, note that for any φ ∈ D, the common domain of all H(s),

(6.48)d

dteitH(s)e−itH(s0)φ = ieitH(s) (Φ(s)− Φ(s0)) e

−itH(s0)φ

from which the well-known Duhamel’s formula is proven. As a consequence,

(6.49)∥

∥

∥

(

e−itH(s) − e−itH(s0))

ψ∥

∥

∥≤∫ t

0

∥

∥

∥(Φ(s)−Φ(s0)) e

−iuH(s0)ψ∥

∥

∥du

is valid for all ψ ∈ H and t ≥ 0 (a similar bound holds for t < 0). Dominated convergence applied

here, using the continuity assumption on Φ, shows that s 7→ eitH(s) is continuous in the strongoperator topology for each fixed t ∈ R. The proof is now completed as described above.

6.3.2. Two particular weighted integrals. For the applications that follow, two particular weightedintegral operators play a key role. We introduce a notation for them here and discuss some basicproperties.

Generally, the set-up is as before. Let H be a densely defined self-adjoint operator on a Hilbertspace H and denote by τt the corresponding dynamics, see e.g. (6.42).


For any fixed γ > 0, let wγ , Wγ ∈ L1(R) be any real-valued functions so that (6.33), (6.38), and(6.39) hold. Define two linear maps F ,G : B(H) → B(H) by setting

(6.50) F(A) =

∫

R

τt(A)wγ(t) dt and G(A) =∫

R

τt(A)Wγ(t) dt.

As we will see, the properties of F and G depend crucially on the choice of γ > 0.In the remainder of this and the next subsection (subsection 6.4) we do not require the more

detailed properties of wγ and Wγ that we have proved for the specific functions constructed inSection 6.2 (see (6.18), (6.32), and (6.34)). These properties will become important later when weanalyze the quasi-locality properties of the spectral flow. In particular, in the following lemma andthe proof of Theorem 6.3, the specific functions defined in Section 6.2 are not required.

Lemma 6.8. Let H be a densely defined, self-adjoint operator on a Hilbert space H. Let γ > 0,wγ ,Wγ ∈ L1(R) be real-valued and satisfy (6.33), (6.38) and (6.39), and F ,G : B(H) → B(H) be asdefined in (6.50). Suppose that the spectrum of H can be decomposed into two non-empty, disjointsets Σ1 and Σ2,

(6.51) spec(H) = Σ1 ∪ Σ2

with Σ1 contained in some compact set and d(Σ1,Σ2) ≥ γ. Denote by P the spectral projectionassociated to H onto Σ1. Then, for any A ∈ B(H)

(6.52) [F(A), P ] = 0

and

(6.53) [G(A), P ] = i

∫

Σ1×Σ2

1

µ− λdEλAdEµ + i

∫

Σ1×Σ2

1

µ− λdEµAdEλ.

Here, Eλ denotes the spectral family associated to H.

Proof. We first prove (6.52). In fact, we will show that each F(A) is diagonal with respect to P inthe sense that

(6.54) PF(A)(1l − P ) = (1l− P )F(A)P = 0 for any A ∈ B(H) .

Given this, one readily checks that

(6.55) [F(A), P ] = (F(A)P − PF(A)P )− (PF(A) − PF(A)P ) = 0

as claimed.We now calculate the left-hand-side of (6.54). To do so, we will use results on double operator

integrals, see e.g. [17]. In fact, using Theorem 4.1 in [17], one sees that

PF(A)(1l − P ) =

∫

R

PeitHAe−itH(1l− P )wγ(t) dt

=

∫

R

∫

Σ1×Σ2

eit(λ−µ)wγ(t)dEλAdEµ dt

=√2π

∫

Σ1×Σ2

wγ(µ− λ)dEλAdEµ = 0 .(6.56)

Here we have used Eλ to denote the spectral family associated to H. Moreover, wγ ∈ L1(R) is suf-ficient to guarantee the re-ordering of the integrals above; it is here that we apply Theorem 4.1 (iii)of [17]. The final equality is due to the fact that the Fourier transform of wγ is supported in [−γ, γ],see (6.33). The other relation in (6.54) is proven similarly, and (6.52) follows.


Arguing as above, we find that

[G(A), P ] = (1l− P )G(A)P − PG(A)(1l − P )

=√2π

∫

Σ1×Σ2

Wγ(λ− µ)dEµAdEλ −√2π

∫

Σ1×Σ2

Wγ(µ− λ)dEλAdEµ(6.57)

The claim in (6.53) now follows from (6.39).

A useful observation for certain applications (see, e.g., [9,84]) is that the map G is a (left-) inverseof the Liouvillean [H, ·] on the space of off-diagonal operators.

Proposition 6.9. Let H be a densely defined, self-adjoint operator on a Hilbert space H, and let[H, ·] denote the generator of the Heisenberg dynamics generated by H. Let γ > 0 and F and G asdefined in (6.50). Suppose that the spectrum of H can be decomposed into two non-empty, disjointsets Σ1 and Σ2, with Σ1 compact and d(Σ1,Σ2) ≥ γ. Let P denote the spectral projection of Honto Σ1. Then, for all A ∈ B(H) such that G(A) ∈ dom[H, ·], we have

(6.58) i[H,G(A)] = F(A)−A.

If, in addition A is off-diagonal with respect to P , meaning A ∈ PB(H)(1 − P )⊕ (1− P )B(H)P ,we have F(A) = 0 and

(6.59) − i[H,G(A)] = A

Proof. For any u ∈ R,

(6.60) τu(G(A)) =∫

R

τt+u(A)Wγ(t) dt =

∫

R

τy(A)Wγ(y − u) dy

Since, by assumption, G(A) ∈ dom[H, ·], and dom[H, ·] is τu-invariant, we then have

i[H, τu(G(A))] =d

duτu(G(A)) =

∫

R

τy(A)d

duWγ(y − u) dy

= F(τu(A)) − τu(A)(6.61)

where the derivative ofWγ is taken in the distributional sense. Evaluation of (6.61) at u = 0 resultsin:

(6.62) i[H,G(A)] = F(A)−A

If A ∈ PB(H)(1 − P ) (or A ∈ (1 − P )B(H)P ), then F(A) ∈ PB(H)(1 − P ) (or F(A) ∈ (1 −P )B(H)P ), and hence in either case, by (6.54), we have F(A) = 0. With this, (6.62) becomes

(6.63) − i[H,G(A)] = A

In applications to quantum spin systems, either finite or infinite, the domain condition on G(A)in this proposition is quite generally satisfied due to the quasi-locality properties of both G and thegenerator of the Heisenberg dynamics See, e.g., the discussion of the domain of the generator ofthe dynamics in the proof of Theorem 7.6.

6.4. The proof of Theorem 6.3. The goal of this section is to complete the proof of Theorem 6.3.Let us recap our progress so far.

Let H(s) = H + Φ(s) be as defined in (6.3). For any γ > 0, a map D : [0, 1] → B(H) is definedby

(6.64) D(s) =

∫

R

τ(s)t (Φ′(s))Wγ(t) dt,

where τ(s)t is the dynamics associated to H(s) as in (6.4) and Wγ is the particular weight function

defined in (6.34). By Lemma 6.7, D(s) is pointwise self-adjoint and continuous in the strong


operator topology. In this case, for any 0 ≤ s ≤ 1, an automorphism αs of B(H) is defined bysetting

(6.65) αs(A) = U(s)∗AU(s) for any A ∈ B(H),

where the unitary U(s) is the unique strong solution of

(6.66)d

dsU(s) = −iD(s)U(s) with U(0) = 1l .

The proof of Theorem 6.3 is completed by showing that if H(s) satisfies Assumption 6.2 for someγ > 0, then the automorphisms αs introduced above satisfy (6.12), i.e.

(6.67) αs(P (s)) = P (0) for all 0 ≤ s ≤ 1 .

Proof of Theorem 6.3: As discussed above, we need only verify (6.67). A formal calculation showsthat

(6.68)d

dsαs(P (s)) = αs

(

i[D(s), P (s)] +d

dsP (s)

)

in the sense of strong derivatives. Since α0(P (0)) = P (0), we need only prove that

(6.69)d

dsP (s) = −i[D(s), P (s)] .

It is well-known, see [67], that spectral projections can be determined through a contour integralof the resolvent, i.e.

(6.70) P (s) = − 1

2πi

∫

η(s)R(z, s) dz,

where R(z, s) = (H(s)− z)−1 is the resolvent of H(s) and η(s) is any contour in the complex planethat encircles the interval I(s), as described in Assumption 6.2. From this representation, it is clearthat strong differentiability of P follows from strong differentiability of R(z, ·), and so the formalcalculation in (6.68) is well-defined. Now, note that for any fixed s0 ∈ [0, 1] the gap assumptionallows for a choice of contour η(s) which is independent of s in a neighborhood of s0. With such acontour one checks that

(6.71)d

dsP (s) =

1

2πi

∫

η(s)R(z, s)Φ′(s)R(z, s) dz.

As P (s) is a strongly differentiable family of orthogonal projections, one can also verify that

(6.72) P (s)d

dsP (s)P (s) = (1l− P (s))

d

dsP (s)(1l− P (s)) = 0.

We conclude that

(6.73)d

dsP (s) =

1

2πi

∫

η(s)A(s, z)Φ′(s)B(s, z) dz +

1

2πi

∫

η(s)B(s, z)∗Φ′(s)A(s, z)∗ dz,

where we have set

(6.74) A(s, z) = P (s)R(z, s) and B(s, z) = R(z, s)(1l − P (s)).

To simplify the integrals on the right-hand-side of (6.73), we again appeal to the formalism of

double operator integrals. In fact, let us denote by, E(s)λ , the spectral family associated to the

self-adjoint operator H(s). One checks that

1

2πi

∫

η(s)A(s, z)Φ′(s)B(s, z) dz =

∫

Σ1(s)

∫

Σ2(s)

1

2πi

∫

η(s)

1

λ− z

1

µ− zdz dE

(s)λ Φ′(s) dE(s)

µ

=

∫

Σ1(s)

∫

Σ2(s)

1

µ− λdE


µ ,(6.75)


where again, the re-ordering of the integrals appearing above is justified by Theorem 4.1 (iii)in [17]. Here specifically, the required integrability condition on the contour is readily verified usingAssumption 6.2. Applying similar arguments to the second term in (6.73), we find that

(6.76)d

dsP (s) =

∫

Σ1(s)

∫

Σ2(s)

1

µ− λdE


µ +

∫

Σ2(s)

∫

Σ1(s)

1

µ− λdE(s)

µ Φ′(s) dE(s)λ .

On the other hand, the right-hand-side of (6.69) is clearly given by

(6.77) − i[D(s), P (s)] = [G(s)(−iΦ′(s)), P (s)] for any 0 ≤ s ≤ 1 .

Here we have used the notation G(s) for the weighted integral operator, see (6.50), defined with

respect to the parameter dependent dynamics, τ(s)t . Using Lemma 6.8, in particular (6.53) with A =

−iΦ′(s), the equality claimed in (6.69) is now clear. This completes the proof of Theorem 6.3.

6.5. Quasi-locality of the spectral flow. For the remainder of this section, let us assume that(Γ, d) is a ν-regular metric space, in the sense of (5.35), and Hx is the complex Hilbert space of thequantum system at x ∈ Γ. We start by considering a finite system corresponding to Λ ∈ P0(Γ).Recall that for any X ⊂ Λ, we denote by HX =

⊗

x∈X Hx and AX = B(HX).This section is divided into two parts. First, in Section 6.5.1, we prove quasi-locality estimates

for the two weighted integral operators introduced in Section 6.3.2. Then, in Section 6.5.2, weestablish quasi-locality bounds for the spectral flow constructed in the proof of Theorem 6.3.

6.5.1. Quasi-locality for two weighted integral operators. In Section 6.3.2, we introduced two par-ticular weighted integral operators that will appear frequently in our applications. We now demon-strate that, under certain additional conditions, each of these weighted integral operators satisfiesan explicit quasi-locality estimate in the sense of Section 5.

Let us assume that there is a one-parameter family of automorphisms of AΛ, which we denote byτt, that satisfies a quasi-locality estimate. More precisely, suppose that there are positive numbersC and v as well as a non-negative, non-decreasing function g for which: given any X,Y ⊂ Λ,

(6.78) ‖[τt(A), B]‖ ≤ C‖A‖‖B‖|X|ev|t|−g(d)

for all A ∈ AX , B ∈ AY , and t ∈ R. Here d = d(X,Y ) is the distance between the sets X and Y .As is discussed in Section 3, such a bound is known for the dynamics generated by a short rangeHamiltonian; it is, e.g., a consequence of the Lieb-Robinson bounds in Theorem 3.1. In order toprove the quasi-local bounds below, we need only know (6.78) and that g(d) becomes sufficientlylarge (see (6.90)). In applications, we typically have (6.78) with

(6.79) limd→∞

g(d) = +∞

In terms of these automorphisms τt, for each γ > 0 define F ,G : B(H) → B(H) by

(6.80) F(A) =

∫

R

τt(A)wγ(t) dt and G(A) =∫

R

τt(A)Wγ(t) dt

for any A ∈ B(H); compare with (6.50). Here again wγ and Wγ are the specific weight functionsintroduced in Section 6.2.

Before we state our first result, recall that wγ(t) = γw(γt) and therefore ‖wγ‖∞ ≤ γc with c theL1-normalization of w, see (6.18). Moreover, Corollary 6.6, see specifically (6.40), demonstratesthat there is an η ∈ (2/7, 1) for which given any x ≥ γ−1e9 the bound

(6.81)

∫ ∞

xwγ(t) dt ≤

27

14ce4fγ(x)

2e−ηfγ (x)


holds. Here, for any b > 0, we have introduced the subadditive, non-decreasing function

(6.82) fb(x) =

e2

4 if 0 ≤ x ≤ b−1e2,bx

ln(bx)2 if x ≥ b−1e2.

See Section 8.2.1 for a discussion of the properties of fb.Our quasi-locality estimate on the weighted integral operator F follows.

Lemma 6.10. Let τt be a family of automorphisms of B(H) satisfying (6.78) with (6.79). Letγ > 0 and take F to be the weighted integral operator defined in (6.80). For any 0 < ǫ < 1 and allX,Y ⊂ Λ the bound

(6.83) ‖[F(A), B]‖ ≤ 2‖A‖‖B‖|X|GǫF (d(X,Y ))

holds for all A ∈ AX and B ∈ AY . Here

(6.84) GǫF (d) =

1 if 0 ≤ d ≤ d∗ǫ

min

1, c(

Cγv + 27

7 e4fγǫ(g(d))

2)

e−ηfγǫ (g(d))

otherwise,

where d∗ǫ is the smallest value of d for which

(6.85) max

[

9,

√

ηγǫǫ

]

≤ ln(γǫg(d)) where γǫ =(1− ǫ)γ

v.

It can be verified that the function GǫF (d) given in (6.84) is monotone and strictly decreasing

when |GǫF (d)| < 1.

Proof. Let X,Y ⊂ Λ. Since wγ is L1-normalized, it is clear that

(6.86) ‖[F(A), B]‖ ≤ 2‖A‖‖B‖ for all A ∈ AX and B ∈ AY .

In applications, this bound is best when d = d(X,Y ) is small.When d = d(X,Y ) is sufficiently large, see below, a different estimate holds. In fact, let T ≥ 0

and estimate

(6.87) ‖[F(A), B]‖ ≤∫

|t|≤T‖[τt(A), B]‖wγ(t) dt+

∫

|t|>T‖[τt(A), B]‖wγ(t) dt .

For the first term above, we ignore the weight and use the locality bound for the dynamics, i.e.(6.78). For the second term, we ignore the dynamics and use the estimate on the weight, see (6.81)above. From these, we obtain the bound

(6.88) ‖[F(A), B]‖ ≤ 2c‖A‖‖B‖|X|(

Cγ

vevT−g(d) +

27

7e4fγ(T )

2e−ηfγ (T )

)

It is important to note that Corollary 6.6, summarized in (6.81) above, has a constraint, and so

(6.88) is only valid if γT ≥ e9. For any 0 < ǫ < 1, choose T = (1−ǫ)v g(d). In this case, we find that

(6.89) ‖[F(A), B]‖ ≤ 2c‖A‖‖B‖|X|(

Cγ

ve−ǫg(d) +

27

7e4fγǫ(g(d))

2e−ηfγǫ (g(d))

)

whenever γǫg(d) ≥ e9 and γǫ is as in (6.85). Since limd→∞ g(d) = ∞, it is clear that (6.89) can beestimated as in (6.84) for sufficiently large d. The relation

(6.90) − ǫg(d) + ηfγǫ(g(d)) = −ǫg(d)[

1− ηγǫǫ ln(γǫg(d))2

]

is used when defining d∗ǫ as above. This completes the proof.


Depending on the application one has in mind, more decay of the function governing the localityof the dynamics, specifically e−g(d), may be needed. For example, many applications require certainmoments of the function Gǫ

F to be finite. Let us make two observations in this regard.On polynomial decay: Let us consider a family of automorphisms τt with a locality estimate of

the form (6.78). If the non-decreasing function g is of the form

(6.91) gq(x) = q ln(1 + x) for some q > 0

then (for fixed t) the locality bound decays like a power-law. In this case, Lemma 6.10 holds,however, the resulting decay function, see (6.84), has no finite moments. In fact, for any positivenumbers a, b, c, and d, one readily checks that

(6.92) limx→∞

(1 + x)ae−bfc(gd(x)) = +∞

where the functions fc and gd are as defined in (6.82) and (6.91) respectively. As we will see in [96],this lack of moments restricts the known proofs of stability of the spectral gap to perturbations thatdecay faster than any polynomial. We do not believe that arguments in [83] can be extended toobtain a uniform lower bound for the spectral gap in the case of perturbations with only power-lawdecay, contrary to the claim made in that work. To see why, note that the proof of Lemma 6.10depends on the choice of T ≥ 0, see e.g. (6.87). This choice must be made in such a way that bothterms on the right-hand-side of (6.88) decay. In order for the first term to decay, vT − g(d) < 0and so one must take T < v−1g(d). As the function fγ(T ) is increasing for large T , the most decayone can obtain from the second term is when T = v−1g(d). If g is logarithmic as discussed above,then even this choice has no finite moments.

On stretched-exponential decay: Let us consider a family of automorphisms τt with a localityestimate of the form (6.78). If the non-decreasing function g is of the form

(6.93) g(r) ≥ arθ for some a > 0 and 0 < θ ≤ 1

then all moments of the decay function, see (6.84), are finite. In fact, for any δ > 0, there is anumber Cδ for which ln(x) ≤ Cδx

δ whenever x ≥ 1. In this case, for any b > 0,

(6.94) fb(g(r)) =bg(r)

ln(bg(r))2≥ C−2

δ (bg(r))1−2δ ≥ (ab)1−2δ

C2δ

rθ(1−2δ)

and therefore, for any δ < 1/2, the function in (6.84) decays at least as fast as a stretched expo-nential, see Section 8.2.1 more on this terminology.

The following lemma is the analogue of Lemma 6.10 applicable to G.Lemma 6.11. Let τt be a family of automorphisms of B(H) satisfying (6.78) with (6.79). Letγ > 0 and take G to be the weighted integral operator defined in (6.80). For any 0 < ǫ < 1 and allX,Y ⊂ Λ the bound

(6.95) ‖[G(A), B]‖ ≤ 2‖A‖‖B‖|X|GǫG (d(X,Y ))

holds for all A ∈ AX and B ∈ AY . Here

(6.96) GǫG(d) =

‖Wγ‖1 if 0 ≤ d ≤ d∗ǫ

min

‖Wγ‖1,(

C2v + 243

49γη ce4fγǫ(g(d))

3)

e−ηfγǫ (g(d))

otherwise,

and d∗ǫ is as defined in Lemma 6.10.

The proof of this lemma is almost identical to that of Lemma 6.10 except that one uses theestimate (6.41) from Corollary 6.6, instead of (6.40). We also use that ‖Wγ‖∞ ≤ 1/2.

Here too, it can be verified that the function GǫG(d) given in (6.96) is monotone and strictly

decreasing when |GǫG(d)| < ‖Wγ‖1.


6.5.2. Quasi-locality of the spectral flow automorphism. We will now consider the spectral flow inthe thermodynamic limit. In order to derive explicit estimates useful for applications, we workwith F -functions of the ν-regular metric space (Γ, d) of the form F (r) = e−g(r)F0(r), where g isnon-decreasing and subadditive, and

(6.97) F0(r) =1

(1 + r)ξ,

for a suitable ξ > 0. As is shown in the Appendix (Section 8.1.1), any choice of ξ > ν + 1 willdefine an F -function on a ν-regular (Γ, d). In the case Γ = Z

ν , ξ > ν is sufficient. We will say thatF is a weighted F -function on (Γ, d) with base F0.

Let us now introduce the models we consider through an assumption.

Assumption 6.12. There is a collection Hxx∈Γ of densely defined, self-adjoint on-site Hamiltoni-ans. For each 0 ≤ s ≤ 1, there is an interaction Φ(s) on AΓ for which

(i) For each X ∈ P0(Γ), Φ(X, s)∗ = Φ(X, s) ∈ AX for all 0 ≤ s ≤ 1.

(ii) For each X ∈ P0(Γ), Φ(X, ·) : [0, 1] → AX strongly C1 in the sense of Definition 6.1.(iii) F is a weighted F -function on (Γ, d) with base F0 as in (6.97) and there is a bounded,

measurable function ‖Φ‖1,1 : [0, 1] → [0,∞) for which given any x, y ∈ Γ, the estimate

(6.98)∑

X∈P0(Γ):

x,y∈X

(

‖Φ(X, s)‖ + |X|‖Φ′(X, s)‖)

≤ ‖Φ‖1,1(s)F (d(x, y))

holds for all 0 ≤ s ≤ 1.

Under Assumption 6.12, given any Λ ∈ P0(Γ),

(6.99) HΛ(s) = HΛ +ΦΛ(s) with HΛ =∑

x∈Λ

Hx and ΦΛ(s) =∑

X⊂Λ

Φ(X, s)

is a well-defined self-adjoint operator on HΛ for each 0 ≤ s ≤ 1. If we denote by Dx ⊂ Hx

the dense domain of the on-site Hamiltonian Hx, then each HΛ(s) has the same dense domainDΛ =

⊗

x∈ΛDx ⊂ HΛ. We stress here that in our applications the number 0 ≤ s ≤ 1 plays the roleof a parameter. In this case, the finite-volume Hamiltonians HΛ(s) do not depend on time t, andthus using functional calculus, for each 0 ≤ s ≤ 1, the dynamics corresponding to these self-adjointoperators is simply given by

(6.100) τΛ,st (A) = eitHΛ(s)Ae−itHΛ(s) for any A ∈ AΛ and t ∈ R .

Interactions depending on the time t itself, as in the model (3.62) discussed in Section 3.2, can beaccommodated without difficulty.

Assumption 6.12 implies that Φ(s) ∈ BF for each 0 ≤ s ≤ 1. Therefore, an application ofTheorem 3.5 shows that there is an infinite volume dynamics defined by

(6.101) τ st (A) = limΛ→Γ

τΛ,st (A) for each A ∈ AlocΓ and t ∈ R .

In terms of this infinite volume dynamics, and any γ > 0, we now define a family of linear mapsKγ

s : AlocΓ → AΓs∈[0,1] by setting

(6.102) Kγs (A) =

∫

R

τ st (A)Wγ(t) dt for any A ∈ AlocΓ .

Here Wγ is the weight function introduced in (6.34) of Section 6.2. Often, we will regard γ as fixedand drop it from our notation.

The following proposition relates the quantities introduced above to the methods discussed inSection 5.4.3.


Proposition 6.13. Consider a quantum lattice system comprised of a ν-regular metric space (Γ, d)

and AΓ. Suppose Assumption 6.12 holds with a weighted F -function F (r) = e−g(r)F0(r) for whichlimr→∞ g(r) = ∞ and F0 is as in (6.97). Then, for any γ > 0, the family of maps Kγ

s s∈[0,1], asdefined in (6.102) above, satisfies the conditions of Assumption 5.15.

Proof. As is clear from the statement, the results we prove below hold for any γ > 0. For conve-nience of presentation, we now fix such a value γ > 0 and then suppress it in our notation.

Let Λnn≥1 be any increasing, exhaustive sequence of (non-empty) finite subsets of Γ. For each

n ≥ 1, define a family of maps K(n)s s∈[0,1], with K(n)

s : AΛn → AΛn for all 0 ≤ s ≤ 1, by setting

(6.103) K(n)s (A) =

∫

R

τΛn,st (A)Wγ(t) dt for any A ∈ AΛn .

Here we have used the finite volume dynamics, see (6.99) and (6.100), with Λ = Λn. We will show

that any such choice of Λnn≥1 determines a sequence of maps K(n)s , as defined in (6.103), which

satisfies all four conditions in Assumption 5.15.The first part of Assumption 5.15 requires we show that for each n ≥ 1, the finite volume families

of maps K(n)s s∈[0,1] satisfy Assumption 5.11. SinceWγ ∈ L1(R) is real-valued, Assumption 5.11 (i)

is easily verified. We note that integrability of Wγ is a consequence of the estimate (6.41) inCorollary 6.6. To check the remaining parts of Assumption 5.11, we recall that properties ofmaps with the form (6.103) were discussed in Example 4.11. Assumption 6.12 guarantees that themethods of Example 4.11 apply, and the remaining details are readily checked.

For our applications, the simple bound

(6.104) ‖K(n)s (A)‖ ≤ ‖Wγ‖1‖A‖

suffices, and thus Assumption 5.15 (ii) is trivially satisfied with p = 0 and B(s) = ‖Wγ‖1.The uniform quasi-locality estimate in Assumption 5.15 (iii) can be seen as follows. By As-

sumption 6.12 (iii), Φ(s) ∈ BF for each 0 ≤ s ≤ 1. As a result, given any n ≥ 1, the model’s

finite-volume dynamics, i.e. τΛn,st , satisfies a Lieb-Robinson bound as in Theorem 3.3. In fact, for

any X,Y ⊂ Λn, with X ∩ Y 6= ∅, and any A ∈ AX and B ∈ AY , the bound

‖[τΛn,st (A), B]‖ ≤ 2‖A‖‖B‖

CF

(

e2CF ‖Φ(s)‖F |t| − 1)

∑

x∈X

∑

y∈Y

F (d(x, y))

≤ C‖A‖‖B‖|X|ev|t|−g(d(X,Y ))(6.105)

holds for all t ∈ R and s ∈ [0, 1]. Here we have estimated the weighted F -function F (r) =

e−g(r)F0(r) and set

(6.106) C =2

CF‖F0‖ and v = 2CF sup

0≤s≤1‖Φ(s)‖F .

In this case, Lemma 6.11 applies. We have that for any 0 < ǫ < 1 and A and B as above, thebound

(6.107) ‖[K(n)s (A), B]‖ ≤ 2‖A‖‖B‖|X|Gǫ(d(X,Y ))

holds with decay function Gǫ as in (6.96). In principle, here we have used that limr→∞ g(r) = ∞,although one only needs that it becomes sufficiently large. Note further that the bound in (6.107)above is uniform in n ≥ 1 as well as 0 ≤ s ≤ 1, and we have proven Assumption 5.15 (iii).

We now demonstrate that Assumption 5.15 (iv) holds. Fix X ∈ P0(Γ) and A ∈ AX . For n ≥ 1sufficiently large, X ⊂ Λn and thus for any T > 0, the following estimate holds:

(6.108) ‖Ks(A)−K(n)s (A)‖ ≤

∫

|t|≤T‖τ st (A)− τΛn,s

t (A)‖|Wγ(t)| dt+ 2‖A‖∫

|t|>T|Wγ(t)| dt.


Appealing to the continuity result in Corollary 3.6, see specifically (3.80), for any t ∈ [−T, T ],

‖τ st (A)− τΛn,st (A)‖ ≤ ‖A‖

CFv|t|ev|t|

∑

x∈X

∑

y∈Γ\Λn

F (d(x, y))

≤ v‖F0‖CF

|X|‖A‖|t|evT−g(d(X,Γ\Λn ))(6.109)

where v is as in (6.106), and in this particular application,

(6.110) 2It,0(Φ(s)) = 2CF ‖Φ(s)‖F |t| ≤ v|t| ,see e.g. (3.21). Thus if T > 0 is sufficiently large (e.g. γT ≥ e9), then by Corollary 6.6,

(6.111)

∫

|t|>T|Wγ(t)| dt ≤

972

49γηce4 (fγ(T ))

3 e−ηfγ(T ) ,

where fγ(t) is as defined in (6.82). Arguing now as in the proof of Lemma 6.10, let 0 < ǫ < 1, setdn = d(X,Γ \ Λn), and take T to be defined by vT = (1− ǫ)g(dn). We have proven that there arepositive numbers C1 and C2 for which

(6.112) ‖Ks(A)−K(n)s (A)‖ ≤ 2‖A‖|X|

(

C1 + C2fγǫ(g(dn))3)

e−ηfγǫ (g(dn))

for n ≥ 1 sufficiently large. For example, one may take

(6.113) C1 =v‖F0‖2CF

∫

R

|t||Wγ(t)| dt and C2 =972

49γηce4 .

This completes Assumption 5.15 (iv), and so Proposition 6.13 is proven.

We can now introduce the spectral flow for the class of models under consideration. As above, allcomments below are valid for any choice of γ > 0, which is suppressed in the notation. The spectralflow automorphism can be defined for any choice of γ > 0 and, under the general assumptions above,the spectral flow is quasi-local with γ-dependent estimates. It is only for the special relation withthe spectral projection P (s) as in (6.12) that γ needs to be a lower bound for the gap in thespectrum as described in Assumption 6.2.

Given Proposition 6.13, we know that the family of maps Kss∈[0,1] satisfies Assumption 5.15.By Assumption 6.12, Φ′ is a well-defined interaction on AΓ, and moreover, Φ′ is a suitable initialinteraction in the sense of Assumption 5.16, in particular, Φ′

1 ∈ BF ([0, 1]) as is clear from (6.98).In this case, we are in a position to apply Theorem 5.17; we first introduce the relevant notation.Let Λnn≥1 be a sequence of (non-empty) increasing and exhaustive finite subsets of Γ. For eachn ≥ 1 and any 0 ≤ s ≤ 1, consider the transformed (bounded) Hamiltonian

(6.114) K(n)s (HΦ′

Λn(s)) =

∑

X⊂Λn

K(n)s (Φ′(X, s))

which acts on HΛn ; compare with (5.95). As in Section 5.4.3, see the unique strong solution of

(6.115)d

dsUn(s) = −iK(n)

s (HΦ′

Λn(s))Un(s) with Un(0) = 1

can be used to define a family of automorphisms of AΛn by setting

(6.116) α(n)s (A) = Un(s)

∗AUn(s) for all A ∈ AΛn and s ∈ [0, 1] ,

see specifically (5.98) and (5.97). We will refer to the automorphisms α(n)s as the finite-volume

spectral flow dynamics. To estimate the quasi-locality of α(n)s , we fix a locally normal product state


ρ on AΓ and proceed as in Section 5.4.3. Consider the finite-volume, s-dependent interaction Ψn

with terms Ψn(Z, s), for Z ⊂ Λn and 0 ≤ s ≤ 1, defined by

(6.117) Ψn(Z, s) =∑

m≥0

∑

X⊂Z:

X(m)∩Λn=Z

∆Λn

X(m)(K(n)s (Φ′(X, s)))

and further, the corresponding infinite-volume interaction Ψ given by

(6.118) Ψ(Z, s) =∑

m≥0

∑

X⊂Z:

X(m)=Z

∆X(m)(Ks(Φ′(X, s))) for any Z ∈ P0(Γ) and each s ∈ [0, 1] ,

again, one should compare with (5.98) and (5.100).The main result of this subsection is as follows.

Theorem 6.14. Consider a quantum lattice system comprised of a ν-regular metric space (Γ, d) and

AΓ. Suppose that Assumption 6.12 holds with a weighted F -function of the form F (r) = e−g(r)F0(r)where F0(r) is as in (6.97) and

(6.119) g(r) ≥ arθ

for some a > 0 and 0 < θ ≤ 1. Then, Ψn converges locally in F -norm to Ψ with respect to anF -function on (Γ, d). Here Ψn and Ψ are as defined in (6.117) and (6.118) above.

We make some comments and point out two corollaries before proving the theorem.First, in principle, one can do better than the growth assumption in (6.119); in fact, one needs

only that there is some 0 < ǫ < 1 for which the decay function Gǫ, see (6.107), satisfies the condi-tions of Theorem 5.17 (ii). As can be seen from our comments in Section 6.5.1 after Lemma 6.10,any weight function g satisfying (6.119) corresponds to such a decay function Gǫ which satisfies theconditions of Theorem 5.17 (ii). However, no weight function g which grows proportional to a loga-rithm, see (6.91), corresponds to a decay function Gǫ satisfying the conditions of Theorem 5.17 (ii).

Next, let us say some more about the F -function whose existence plays a crucial role in the proofof Theorem 6.14. Given the decay of the initial interaction Φ, see (6.98), it is proven in Proposi-tion 6.13 that the weighted integral operators used in defining the generator of the spectral flow,see (6.103) and subsequently (6.114), satisfy the quasi-locality estimate (6.107). The correspond-ing decay function G (take ǫ = 1/2 for convenience) has the following form: there exist positivenumbers C1, C2, and d∗ for which

(6.120) G(d) ∼

C1 0 ≤ d ≤ d∗,

C2 exp[

−ηf γ2v(g(d))

]

d > d∗

where we stress that the function g above corresponds to the weight in the F -function governingthe decay of the initial interaction Φ. It should be clear that any F -function governing the decay ofΨ (and similarly Ψn) will decay no faster than this G. Our estimates show that: there are positivenumbers C ′

1, C′2, and d

′∗ for which Ψ,Ψn ∈ BF ([0, 1]) with

(6.121) F (d) =

C ′1 0 ≤ d ≤ d′∗C′

2

(1+d)ξexp

[

−η′f γ2v(g(d))

]

d > d′∗

where η′ is any number strictly less that η and the function g may be taken as g(d) = adθ withthe same value of θ (as g) but, in general, a smaller value of a. As is discussed in Section 8.2, anyfunction with the form (6.121) is an F -function on (Γ, d). To obtain the local convergence of Ψn

to Ψ, we will need to modify the above F -function slightly, but all relevant estimates on Ψ andΨn, see for example Corollary 6.15 and Corollary 6.16 below, will be made with respect to thefunction F described above. For more details on this, see the discussion following the statement ofTheorem 5.17.


Finally, our proof of Theorem 6.14 guarantees that Theorem 3.8 applies and so

(6.122) limn→∞

α(n)s (A) = αs(A) for any A ∈ Aloc

Γ and s ∈ [0, 1] .

Here the limit is in norm and the quantity, αs, is the well-defined, infinite-volume dynamics asso-ciated to Ψ ∈ BF ([0, 1]), with terms as in (6.118), whose existence is guaranteed by Theorem 3.5.

For ease of later reference, we now state two corollaries providing explicit estimates on thequasi-locality of the spectral flow.

Corollary 6.15. Under the assumptions of Theorem 6.14, for any X,Y ∈ P0(Γ) with X ∩ Y = ∅,the bound

(6.123) ‖[αs(A), B]‖ ≤ 2‖A‖‖B‖CF

(

e2Is,0(Ψ) − 1)

∑

x∈X

∑

y∈Y

F (d(x, y))

holds for any A ∈ AX , B ∈ AY , and 0 ≤ s ≤ 1. Here F may be taken as in (6.121).

Since Ψ ∈ BF ([0, 1]), the above is an immediate consequence of Corollary 3.6 (i). Our estimatesactually show that sup0≤s≤1 ‖Ψ‖F (s) <∞ and, therefore, we have

(6.124) Is,0(Ψ) = CF

∫ s

0‖Ψ‖F (r) dr ≤ sCF sup

0≤s≤1‖Ψ‖F (s) .

Given (6.124), it is clear that the bound in Corollary 6.15 may be further estimated with a lineardependence on s. This observation is useful in some applications.

The following corollary is a direct application of Theorem 3.8 (i).

Corollary 6.16. Under the assumptions of Theorem 6.14, for any X ∈ P0(Γ), the bound

(6.125) ‖αs(A)−A‖ ≤ 2|X|‖A‖‖F ‖∫ s

0‖Ψ‖F (r) dr

holds for all A ∈ AX and 0 ≤ s ≤ 1. Here F may be taken as in (6.121).

Proof of Theorem 6.14: It is clear that, under the decay assumptions (6.119), Proposition 6.13holds. In this case, for each γ > 0, the family of maps Kγ

ss∈[0,1] satisfies Assumption 5.15, andmoreover, Φ′ is a suitable initial interaction in the sense of Assumption 5.16. As before, we willsuppress the dependence on γ > 0 in what follows.

For any x, y ∈ Γ and n ≥ 1 large enough so that x, y ∈ Λn, an application of Theorem 5.13 shows

(6.126)∑

Z⊂Λn:x,y∈Z

‖Ψn(Z, s)‖ ≤ C1F (d(x, y)/3) +C2

∞∑

m=⌊d(x,y)/3⌋

(1 +m)ν+1Gǫ(m)

where Ψn is the finite-volume interaction in (6.117), F is the weighted F -function governing thedecay of Φ as in Assumption 6.12 (iii), and Gǫ is the decay function associated to the family

K(n)s s∈[0,1] as in (6.107), see also (6.96). Our estimates show that one may take

(6.127) C1 =

(

‖Wγ‖1 + 8κ2∞∑

m=0

(1 +m)2ν+1Gǫ(m)

)

sup0≤s≤1

‖Φ‖1,1(s)

and C2 = 8κ‖F‖ sup0≤s≤1 ‖Φ‖1,1(s). As we have argued before, the analogue of (6.126) also holdswith the interaction Ψ replacing Ψn on the left-hand-side and the right-hand-side unchanged.

It is also clear that the decay assumption (6.119) guarantees that for any 0 < δ < η,

(6.128) Cδ =∞∑

m=0

(1 +m)ν+1+ξ

(

C

2v+

243

49γηce4fγǫ(g(m))3

)

e−δfγǫ (g(m)) <∞


As such, for d = d(x, y) sufficiently large, we may further estimate the right-hand-side of (6.126)by

(6.129) C1e−g(d/3)

(1 + d/3)ξ+ C2Cδ

e−(η−δ)fγǫ (g(⌊d/3⌋))

(1 + ⌊d/3⌋)ξ .

For large d, the second term above dominates. Using the facts provided in Section 8.3, it is clearthat an F -function, F , of the form in (6.121) bounds this quantity. For any such F , our estimatesare uniform with respect to s, and so we have proven that

(6.130) sup0≤s≤1

‖Ψ‖F (s) <∞ .

In fact, the same uniform estimate also holds for the finite-volume interactions Ψn.Since the form of the decay function Gǫ, as in (6.107), is explicit, see (6.96), it is clear that√Gǫ has a finite 2ν + 1 moment. Arguing as above, a similar, yet different, F -function F can be

produced which satisfies the assumptions of Theorem 5.17 (ii) with α = 1/2; in fact, any choice of0 < α < 1 suffices. In this case, Ψn converges locally in F -norm to Ψ with respect to this functionF . As indicated previously, for the estimates in Corollary 6.15 and Corollary 6.16, one can use theoriginal F -function F having the form (6.121).

7. Automorphic equivalence of gapped ground state phases

7.1. Uniformly gapped curves and automorphic equivalence. In this section, we use thespectral flow to study gapped ground state phases of a quantum lattice model (Γ, d) and AΓ.As in the previous sections, we will discuss both finite and infinite volume systems and take thethermodynamic limit along a sequence of increasing and absorbing finite volumes. To this end, wewill consider the following set up for this section.

Throughout this section, let (Γ, d) be a fixed ν-regular metric space with a weighted F -functionof the form F (r) = e−g(r)F0(r), where F0 is an F -function for (Γ, d) of the form (6.97) and g isa non-negative, non-decreasing, subadditive function bounded below by arθ for some θ ∈ (0, 1].In addition, we consider a fixed sequence of increasing and absorbing finite volumes Λn ↑ Γ, andwith the convention that we always take the thermodynamic limit with respect to a subsequenceof this sequence. We will use the notation B1

F ([0, 1]) to denote the space of differentiable curvesof interactions Φ(s) ∈ BF , s ∈ [0, 1], satisfying Assumption 6.12. At each x ∈ Γ we may have adensely defined self-adjoint Hx, but these we regard as fixed. Specifically, we only consider hererelations between models with different interactions Φ(s) but with the same Hx | x ∈ Γ.

For simplicity we will assume that the finite-volume Hamiltonians for the models parametrizedby s ∈ [0, 1], are defined by

(7.1) HΛ(s) =∑

x∈Λ

Hx +∑

X⊂Λ

Φ(X, s).

Within the context described above, we now introduce the notion of a uniformly gapped curveof models or, equivalently, a curve of uniformly gapped interactions for which we use the notationEΛ(s) = inf specHΛ(s) to denote the ground state energy of HΛ(s).

Definition 7.1. Let γ > 0. A curve of interactions Φ ∈ B1F ([0, 1]) is called uniformly gapped with

gap γ, if there exists a non-negative sequence (δn)n, with limn δn = 0, such that for all n ≥ 1 ands ∈ [0, 1]

(7.2) specHΛn(s) ⊂ [EΛn(s), EΛn(s) + δn] ∪ [EΛn(s) + δn + γ,∞),

where HΛn(s) is the finite volume Hamiltonian defined in (7.1).


We can leave γ unspecified and call the curve simply uniformly gapped if there exists γ > 0 suchthat it is uniformly gapped with gap γ.

It is well-known that the spectral gap generally depends on the boundary conditions. Our choiceto define the path of Hamiltonians (7.1) with respect to a single interaction leads to boundaryconditions that are not necessarily the most general ones of interest; studying all possible cases atonce would lead to quite onerous notation, which we want to avoid in this discussion. Suffice it tonote that everything in this section could be generalized to the situation where we have boundaryconditions expressed by a sequence Φn ∈ B1

F ([0, 1]), by requiring that Φn converges locally (in auniform version of Definition 3.7) in a suitable norm to some Φ ∈ B1

F ([0, 1]). In this case, Φn isthen used to define the Hamiltonian (7.1) on the volume Λn for each n. For example, this can beused to study finite systems in Z

ν with periodic boundary conditions. Even without consideringn-dependent interactions, the present set-up allows one to study the effects of certain boundaryconditions. For example, by replacing Γ by a subset Γ0 ⊂ Γ and different sequences of finite volumesΛn, models defined with the same interaction Φ may show different behavior. An example of thisis discussed in detail for a class of so-called PVBS models in [11,18]. There, Γ0 is the half-space inΓ = Z

ν defined by an arbitrary hyperplane. For these models, the spectral gap is shown to dependnon-trivially on the orientation of the hyperplane.

We use the notion of a uniform gap to define a relation ∼ on BF as follows.

Definition 7.2. For Φ0,Φ1 ∈ BF , we say that Φ0 and Φ1 are equivalent, denoted by Φ0 ∼ Φ1, ifthere exists a uniformly gapped curve Φ ∈ B1

F ([0, 1]) such that Φ(0) = Φ0 and Φ(1) = Φ1.

In the physics literature, two models Φ0 and Φ1 are said to be in the same gapped ground statephase if Φ0 ∼ Φ1 [32,33]. Studying curves of models has proved to be fruitful also in mathematicalstudies [12–16]. In this section we explore some essential properties of models that belong to thesame gapped ground state phase. First, however, we show that the relation ∼, used to define thisnotion is indeed an equivalence relation.

Proposition 7.3. The relation ∼ defined in Definition 7.2 is an equivalence relation on BF .

Proof. The defining properties of reflexivity and symmetry of an equivalence relation follow byconsidering constant curves Φ(s) = Φ0, for all s ∈ [0, 1], and reversed curves Φ−1(s) = Φ(1 − s).For transitivity, consider two curves Φ(1)(s), Φ(2)(s) ∈ B1

F ([0, 1]) such that Φ(1)(1) = Φ(2)(0), anddefine Φ(s) ∈ B1

F ([0, 1]) by

Φ(s) =

Φ(1)(sin(πs)) s ≤ 1/2

Φ(2)(1− sin(πs)) s > 1/2

Here, the re-parameterization of s in the piecewise definition is chosen only to ensure the differentia-bility of Φ at s = 1/2. Other re-parameterizations will also work. Transitivity follows from setting

δn = max(δ(1)n , δ

(2)n ), and γ = min(γ(1), γ(2)), where δ

(i)n and γ(i), i = 1, 2, refer to the sequences

and the gap for the two curves.

Note that, without loss of generality, we can assume that the sequence (δn) in Definition 7.1 isnon-increasing. It is also easy to see that for uniformly gapped Φ, the spectral projection Pn(s)of HΛn(s) associated with the interval [EΛn(s), EΛn(s) + δn] becomes independent of the choice ofsequence (δn) for large n in the sense that for any two sequences (δn) and (δ′n) for which (7.2) holds,the spectral projections associated with the intervals [EΛn(s), EΛn(s)+δn] and [EΛn(s), EΛn(s)+δ

′n]

coincide for sufficiently large n.Let Φ be uniformly gapped. Then, consider the collection of states of AΛn supported on the

spectral subspace of HΛn(s) associated with the intervals [EΛn(s), EΛn(s) + δn]. More precisely,define

Sn(s) = ω ∈ S(AΛn) | ω(Pn(s)) = 1.


Here, for any complex C∗-algebra A with unit 1, S(A) denotes the state space of A, that is theset of positive linear functionals on A with ω(1) = 1. The remarks in the previous paragraph showthat Sn(s) becomes independent of the choice of the sequence (δn) for large n. Therefore, it ispossible to define

S(s) = ω ∈ S(AΓ) | ∃(nk) increasing and ωk ∈ Snk(s) s.t. lim

kωk(A) = ω(A),∀A ∈ Aloc

Γ .

In this sense, S(s) is the set of all weak−∗ limits of states in Sn(s). It can be shown that the elementsω ∈ S(s) are ground states of the infinite-volume model defined by the dynamics τt obtained asthe thermodynamic limit of the model with Hamiltonians (7.1) [96]. The prime example to keepin mind is that the set Sn(s) also consists of ground states for a Hamiltonian HΛn(s) that hasa uniform lower-bound, denoted by γ > 0, separating the ground state energy from the rest ofthe spectrum. However, it will be interesting to consider the slightly more general set-up we haveintroduced above.

Let us now fix a uniformly gapped curve Φ ∈ B1F ([0, 1]) with gap γ. As an application of Theo-

rem 6.14 and the comments following it, we have strongly continuous spectral flow automorphisms

α(n)s for the curve of finite-volume on Λn, and αs for the infinite system on Γ. Here, the uniform

gap of the curve plays the role of the parameter γ in the construction of the spectral flow. More-

over, Theorem 6.14, see specifically (6.122), establishes that αs is the strong limit of α(n)s , and the

convergence of this limit is uniform for s ∈ [0, 1]. Moreover, we can use the spectral flow αs toconstruct a co-cycle of automorphisms αt,s := α−1

s αt, for all t, s ∈ [0, 1]. We can similarly define

a collection of finite volume co-cycles, α(n)t,s .

Our next goal is to show that the spectral flow co-cycle establishes a close relationship betweenthe sets S(s) for different values of s, which we refer to as automorphic equivalence of gapped ground

state phases. Using the definition of α(n)t,s above, Theorem 6.3 establishes the following relationships

between the spectral projections Pn(s):

(7.3) α(n)t,s (Pn(t)) = Pn(s), for all s, t ∈ [0, 1].

As an immediate consequence we have that ωn α(n)t,s ∈ Sn(t) for any ωn ∈ Sn(s) as

ωn α(n)t,s (Pn(t)) = ωn(Pn(s)) = 1.

Since α(n)t,s is an automorphism, it is invertible. In fact, its inverse is given by α

(n)s,t . As such, we see

that composition with α(n)t,s defines a bijection between the sets Sn(s) and Sn(t). Explicitly

(7.4) Sn(t) = ωn α(n)t,s | ωn ∈ Sn(s).

The next theorem extends this bijection to the thermodynamic limit. The quasi-local propertiesof the spectral flow established in Section 6 play an important role both at the technical andthe conceptual level. Technically, they are the main ingredient in establishing the convergence ofthe thermodynamic limit. Conceptually, the fact that αs is a dynamics generated by a short-rangeinteraction shows that local properties of the states at different values of s are related by a ‘natural,’finite-time, unitary evolution.

Theorem 7.4. For all s, t ∈ [0, 1], the spectral flow automorphism αs,t provides a bijection betweenthe sets S(s) and S(t) by composition:

(7.5) S(t) = S(s) αt,s

Proof. This is a direct consequence of (7.4), Theorem 6.14 and the Lemma 7.5 below.

Lemma 7.5. Let (αn)n be a strongly convergent sequence of automorphisms of a C∗-algebra A,converging to α and let (ωn)n be a sequence of states on A. Then the following are equivalent:


(i) ωn converges to ω in the weak-∗ topology;(ii) ωn α converges to ω α in the weak-∗ topology;(iii) ωn αn converges to ω α in the weak-∗ topology.

Proof. (i)⇔(ii) follows immediately from the fact that α and α−1 are automorphisms. Now if (ii)holds, the limit of the second term in the RHS of

|(ωn αn)(A)− (ω α)(A)| ≤ |ωn(αn(A)− α(A))| + |ωn(α(A)) − ω(α(A))| ,vanishes. So does the limit of the first term since

|ωn(αn(A)− α(A))| ≤ ‖ωn‖‖αn(A)− α(A)‖ −→ 0

as ωn is a state. Therefore, (iii) holds. A similar argument implies (iii)⇒(ii).

Recall the following essential property of the spectral flow automorphisms with parameter γ > 0constructed in Section 6 for a family of Hamiltonians HΛ(s). Suppose that P (s) is the spectralprojection of HΛ(s) associated to a bounded interval [a(s), b(s)] (with a(s) and b(s) differentiable)that is gapped from the rest of the spectrum by γ > 0, i.e.

(a(s)− γ, a(s)) ∩ spec(HΛ(s)) = (b(s), b(s) + γ) ∩ spec(HΛ(s)) = ∅ for all s ∈ [0, 1].

Then by Theorem 6.3 the spectral flow αt,s with parameter γ associated with HΛ(s) once againmaps P (t) to P (s). In the discussion above we focused on gapped ground state phases, for whichthe relevant part of the spectrum is at the bottom. We will describe examples in some detail inSection 7.3. There is no reason, however, why a similar strategy could not be employed to studystates supported in spectral subspaces associated with an isolated part elsewhere in the spectrum.For example, an isolated band of excited states could also be studied with the help of spectralautomorphisms. This new and largely unexplored territory seems promising to us.

We conclude this section with the following result regarding the continuity of the spectral gapabove the ground state energy of the GNS Hamiltonian Hωs of an infinite volume ground stateωs ∈ S(s) for the case of quantum spin systems, i.e., the single-site Hilbert spaces Hx are finite-dimensional and the dynamics is generated by an interaction Φ ∈ B1

F ([0, 1]) for a suitable F -functionF . The restriction to the case of quantum spin systems is because we rely on some well-knownproperties of the dynamics and, in particular, its generator in that case. The finite-dimensionalityof the single-site Hilbert spaces is not essential, but the boundedness of the interactions is, inaddition to the general setup described at the beginning of this section, including Assumption 6.12.

Theorem 7.6. Consider a quantum spin model defined by a uniformly gapped curve of interac-tions Φ ∈ B1

F ([0, 1]), with gap γ > 0. Fix s0 ∈ [0, 1] and let Hωss∈[0,1] denote the set of GNSHamiltonians associated to the states ωs ∈ S(s) of the form ωs = ωs0 αs,s0 for some ωs0 ∈ S(s0),and with the spectral flow αs,s0 corresponding to the parameter γ. If for all s ∈ [0, 1], kerHωs isone-dimensional, then

γ(s) := supδ > 0 : spec(Hωs) ∩ (0, δ) = ∅is a upper-semicontinuous function of s.

Proof. Recall that for each s, the infinite volume dynamics τ(s)t is a strongly continuous group of

automorphisms of AΓ generated by a closed operator δ(s), i.e. τ(s)t = eitδ

(s), and that Aloc

Γ is a core

for δ(s).First, we show that for all t, s, s0 ∈ I, αt,s(Aloc

Γ ) is a core for δ(s0). Since AlocΓ is a core and

αt,s is an automorphism, we only need to show that αt,s(A) ∈ dom(δ(s0)), for all A ∈ AlocΓ . Let

X = supp(A) and ΠX(n) be as in (4.11). Then by (4.13)

αt,s(A) = limn→∞

ΠX(n)(αt,s(A)).


The result follows from showing that δ(s0)(ΠX(n)(αt,s(A))) is convergent. Using the telescopic

property of ∆X(n) and linearity of δ(s0), it follows that for any n ≥ 0

δ(s0)(ΠX(n)(αt,s(A))) =n∑

m=0

δ(s0)(∆X(m)(αt,s(A))),

which is absolutely convergent by Proposition 5.9.Now, let γ(s) denote the spectral gap of Hωs and pick any s0 ∈ [0, 1]. By the variational principle,

(7.6) γ(s) = infωs(A∗δ(s)(A)) | A ∈ C, ωs(A) = 0, ωs(A

∗A) = 1,where C is any core for δ(s). Using that ωs = ωs0 αs,s0 and Cs = αs0,s(Aloc

Γ ) is a core for δ(s), wehave the following identity:

(7.7) A ∈ Cs | ωs(A) = 0, ωs(A∗A) = 1 = αs,s0(A) | A ∈ Aloc

Γ , ωs0(A) = 0, ωs0(A∗A) = 1.

For any A ∈ AlocΓ , define the function fA(s) = ωs0(A

∗αs,s0 δ(s) αs0,s(A)), and consider the familyof functions,

F = fA | A ∈ AlocΓ , ωs0(A) = 0, ωs0(A

∗A) = 1.Using (7.7), the expression for the gap in (7.6) can be rewritten in terms of the family F as

(7.8) γ(s) = inff(s) | f ∈ F.The result follows from showing that all f ∈ F are continuous. This can be seen by expressing

the operator αs,s0 δ(s) αs0,s as the generator of the dynamics for a new s-dependent interactionΨ ∈ BF (I). Using the continuity of automorphisms and αs,s0 αs0,s = id, we have

(7.9) αs,s0 δ(s) αs0,s(A) = limn[αs,s0(HΛn(s)), A].

Theorem 5.13 and Corollary 6.15 imply the existence of an F -function F , see (5.89), and a stronglycontinuous Ψ ∈ BF (I) such that the RHS is the generator determined by Ψ(s), which is again

locally bounded and quasi-local. It follows that the map s 7→ αs,s0 δ(s) αs0,s(A) is continuous for

each finite X ⊂ Γ and A ∈ AX . Therefore, f(s) = ωs0(A∗αs,s0 δ(s) αs0,s(A)) defines a continuous

function.

As far as we are aware, for all models that satisfy the conditions of the theorem, the gap appearsto be continuous in the parameter, not just semicontinuous. In particular, the gap is continuouswhen perturbation theory applies. This raises the question whether one indeed has continuity ofthe spectral gap as long as it is strictly positive, or whether additional assumptions are neededfor continuity. Needless to say, the gap is not always stable and so should not be expected to becontinuous in general on a domain where it vanishes at some points.

7.2. Automorphic equivalence with symmetry. In the previous section we introduced theclassification of gapped ground state phases through equivalence classes of interactions for whichthere exists an interpolation by a uniformly gapped curve. We showed that within each equiva-lence class the sets of ground states are mapped into each other by an automorphism with goodquasi-locality properties (the spectral flow derived from the uniformly gapped curve of interactionsinterpolating between the models). Implicit in this description is the idea that any curve of in-teractions interpolating between two models in distinct phases (different equivalence classes) mustcontain at least one point where the gap vanishes. Such points are called quantum critical pointsand one says that a quantum phase transition occurs in the system [115].

Physical systems often have symmetries that play an important role. In the description of certainphenomena, it may be essential that a certain symmetry be present in the model. This led to theconcept of symmetry protected gapped phases [31,39] due to the observation that if one only allowscurves of interactions that all possess a given symmetry, a finer classification of gapped ground


state phases may arise. A nice example of this are the Z2 × Z2 protected phases of the spin-1chain [33, 102, 104, 122]. In general, the equivalence classes break up into subclasses if a restrictedset of uniformly gapped curves of interactions is used to define the equivalence relation. Thisprompts us to revisit the notion of automorphic equivalence in the presence of symmetry.

A symmetry is usually specified by the action of a group G (as automorphisms) on the algebra ofobservables of the system. Although there are interesting symmetries that do not fit the frameworkof group representations by automorphisms, such as dualities and quantum group symmetries, welimit the discussion here to that setting. In general, we use the label G to specify the presence of acertain symmetry. So, we will consider spaces of interactions BG

F ⊂ BF and of curves of interactions

B1,GF ([0, 1]) ⊂ B1

F ([0, 1]). To be clear, in this context G stands for the full specification of thesymmetry including its action on the system, not just the abstract group.

Here are four important classes of symmetries:

(i) Local symmetries are described by automorphisms β of AΓ with the property that they leavethe single-site algebras, Ax = B(Hx), invariant. Specifically, we assume the restrictionsof β to Ax, x ∈ Γ, are inner automorphisms given in terms of a unitary Ux ∈ B(Hx):β(A) = U∗

xAUx for A ∈ B(Hx). This type of symmetries are sometimes called gaugesymmetries because gauge symmetries are of this form. Thus, any local symmetry β isdetermined by a family of unitaries Ux ∈ B(Hx) | x ∈ Γ. We say that β is a symmetry ofΦ if β(Φ(X)) = Φ(X), for all X ∈ P0(Γ). It is easy to see that this implies that β commuteswith the dynamics τt generated by Φ: β τt = τt β. If Φ depends on a parameter s oron the time t, the symmetry condition is assumed to hold pointwise in s and/or t. The setof all local symmetries form a group under the law of composition of automorphisms. It isoften useful to consider the (projective) representations of this group, G, given by the localunitaries Ux(g), g ∈ G.

(ii) Lattice symmetries are, in general, described by a bijection R : Γ → Γ. It is usuallyimportant that R preserves the local structure of (Γ, d), e.g., one requires that R is isometric:d(R(x), R(y)) = d(x, y), x, y ∈ Γ. Examples include translations of lattices such as Zν , andreflection symmetries satisfying R2 = id. If we assume that HR(x)

∼= Hx, R can be lifted toan automorphism of AΓ as follows. Denote by ix : B(Hx) → Ax the natural isomorphism(or a well-chosen one) and define the automorphism βR of AΓ, by putting

(7.10) βR(A) = iR(x) i−1x (A), for all A ∈ Ax and x ∈ Γ.

The symmetry of the interaction is expressed by the property Φ(R(X)) = βR(Φ(X)). Inthe case of lattice translations this yields a representation of (Zm,+) on AΓ, i.e., for a ∈Zm, R(x) = x + a denotes the action of translations on Γ, and X + a = x + a | x ∈ X.

Correspondingly, Φ is called translation invariant if βa(Φ(X)) = Φ(X + a), for all a ∈ Zm.

(iii) Time-reversal symmetry is expressed as a local symmetry (discussed in (i)) given by an anti-automorphism, implemented on each site by an anti-unitary transformation. The latter are,in general, the composition of a unitary transformation and a complex conjugation. Besidestaking into account the anti-linearity, time reversal symmetry can be treated in the sameway as linear local symmetries.

(iv) Chiral symmetry is described by a unitary, say C, that anti-commutes with the Hamiltonian.So, at each point in the curve of Hamiltonians we have C∗H(s)C = −H(s). For the dynamicsthis implies that for all s ∈ [0, 1], t ∈ R, and A ∈ A,

(7.11) C∗τH(s)t (A)C = τ

−H(s)t (C∗AC) = τ

H(s)−t (C∗AC).

It should be noted that the basic types of symmetries can be combined. For example, somemodels are invariant under a combined lattice reflection and time-reversal transformation, withoutpossessing either of these symmetries separately.


We assume the same setup as described in the beginning of Section 7 of a fixed ν-regular metricspace (Γ, d) with a specified weighted F -function F . Let G denote the symmetries under consider-ation. The fixed family of on-site Hamiltonians Hx, x ∈ Γ, is assumed to have the symmetry G asif it were a zero-range interaction. For example, if Ux ∈ B(Hx) describes a local unitary symmetry,we assume that the domain of Hx is invariant under Ux and that Hx and Ux commute. Or, asanother example, if Γ = Z

ν and the symmetry is the full translation invariance of the lattice, Hx

is assumed to be the same self-adjoint operator at each site x.For the interactions, let BG

F ⊂ BF denote the space of interactions with finite F -norm that possessthe symmetry G, and

(7.12) B1,GF ([0, 1]) = Φ ∈ B1

F ([0, 1]) : Φ(s) ∈ BGF for all s ∈ [0, 1]

Definitions 7.1 and 7.2 of uniformly gapped curves and the equivalence relation now carry over thesituation with a symmetryG in the obvious way, as does the proof of the analogue of Proposition 7.3.The resulting equivalence classes are called symmetry protected phases. Since the uniformly gappedcurves with symmetry are a special case of the general situation, Theorem 7.4 applies and thespectral flow automorphism establishes a bijection between the sets of states S(s) along the curve.

For the study of the stability of gapped ground state phases with symmetry breaking we presentin our subsequent work [96], it will be important that the automorphisms αt,s commute with theautomorphisms βg, g ∈ G, representing the symmetry on AΓ. Moreover, it will be desirable thatthe interaction Ψ(s) generating αt,s and its finite-system analogues all have the symmetry. Thereare a few subtleties that merit further discussion concerning the construction of a spectral flowwith the desired symmetry properties.

As mentioned above, it is important that both the spectral flow αt,s and its generating interactionΨ(s) respect the symmetry of the initial interaction Φ(s). Recall that the conditional expectationsΠX from (4.11) play a crucial role in the quasi-locality properties of αt,s and the definition of Ψ(s),see Corollary 6.15, (6.118) and Section 4.2. In the presence of a local symmetry β, it is useful tochoose the locally normal product state in the definition of the conditional expectations ΠX thatis β-invariant, meaning ρx(A) = ρx(β(A)), A ∈ Ax or, equivalently, β(ρx) = ρx, see (4.8). Thisrequirement guarantees that if A is invariant under β, then so is ΠX(A), i.e.

(7.13) β(ΠX(A)) = ΠX(β(A)) = ΠX(A).

If dimHx < ∞, then a β-invariant locally normal product state always exists. For example,setting ρx to be the tracial state will produce a β-invariant state. Given any Φ(s) with a localsymmetry and any symmetric ρ, it is easy to see using (7.13) that the Hastings interactions Ψn

defined in (6.117) and, consequently, the spectral flow α(n)s derived from Φ(s) both inherit this

symmetry. The same holds true for the corresponding infinite volume objects, Ψ and αs. Inparticular αs commutes with any local symmetry automorphism β that leaves Φ′(s) invariant forall s ∈ [0, 1].

For infinite-dimensional Hx, a symmetric normal state on Hx may or may not exist. One mayhave to relax either the normality or the symmetry requirement. Which of the two is more relevantwould depend on the situation at hand but for the type of applications we are considering here, it isimportant to use normal states. In the case of a gauge symmetry described by a compact Lie group,constructing symmetric normal states is not a problem. However, even when such a state does notexist, the symmetry of the spectral flow is restored in the thermodynamic limit. This follows fromthe observation that although the infinite-volume interaction Ψ(s) depends on the choice of thelocally normal state ρ used in its construction, the infinite-volume flow is the thermodynamic limitof automorphisms generated by self-adjoint operators that commute with the symmetry. This isapparent, e.g., from the expression (5.71) in which HΦ

Λ is to be replaced by H ′Λ(s), and K is Gs

defined in (6.50).


In the presence of a lattice symmetry such as translation invariance, it makes sense to pick atranslation invariant product state ρ to define the conditional expectations ΠX . This is obviouslyalways possible and yields a covariant family of conditional expectations, meaning that βa ΠX =ΠX+a βa, where for a ∈ Z

m, X + a denotes the action of the translations on Γ and βa denotesthe corresponding action on AΓ. Finite subsystems defined on quotient lattices, e.g. Z

n/(LZm)with n ≥ m, can have the corresponding quotient symmetry (Zm mod L), which is equivalent toconsidering the system with periodic boundary conditions. In general, finite systems will not havean exact translation symmetry but, again, the symmetry is recovered in the thermodynamic limit.

The case of time-reversal symmetry can be treated in the same way as local unitary symmetries.Due to the oddness of the function Wγ in (6.35) the Hastings interaction Ψ(s) changes sign undertime-reversal. Since the time-reversal automorphism is anti-linear, however, this is exactly the

requirement for α(n)s and αs to commute with it.

The case of a fixed chiral symmetry C along the curve of Hamiltonians H(s) = H+Φ(s), impliesthat Φ′(s) anti-commutes with C. Using again the oddness of the function Wγ , and the property(7.11), it is straightforward to check that C then commutes with the generator of the spectral flow,i.e., it is a symmetry of the spectral flow.

Theorem 7.7. Let βg | g ∈ G be the automorphisms on AΓ representing symmetries of thesystem of the type described in (i-iv) above. Then, for any uniformly gapped curve of interactions

Φ ∈ B1,GF ([0, 1]), there exists a strongly continuous co-cycle of spectral flow automorphisms αt,s, s, t ∈

[0, 1], such that

(7.14) S(t) = S(s) αt,s,

and

βg αt,s = αt,s βg, for all s, t ∈ [0, 1], g ∈ G.

The list of types of symmetries we have discussed here is not exhaustive. For example, anothertype of symmetry relevant for applications are duality symmetries. We postpone the discussion ofthose to [96], where we will study the stability of gapped ground state phases.

7.3. Examples of uniformly gapped curves. The construction of the automorphisms αt,s as-sumes the existence of a uniform lower bound for the spectral gap above the ground states alongthe curve of models in the sense of Definition 7.1. Establishing a uniform bound for the gap isgenerally a very hard problem. Fortunately, there are a good number of interesting examples wherethe existence of a positive uniform lower bound can be proved.

The largest variety of examples is found as a result of perturbing models for which the groundstate and the existence of a spectral gap above it are known. We will review the state of the artof perturbative results of this type in [96]. For this reason, we limit ourselves here to citing afew works that illustrate the broad range of examples that exist in the literature: some exactlysolvable models such as the anisotropic XY chain [74], quantum perturbations of classical spinmodels [69,80], perturbations of the AKLT chain [2,128] and similar models [121], perturbations ofsimple models with topological order in the ground state such as the Toric Code Model [21], generalperturbations of frustration-free models satisfying a Local Topological Order Condition [83], andperturbations of quasi-free Fermion systems [37].

Other interesting examples for which explicit lower bounds for the gap can be obtained andclasses of models for which the equivalence classes can be explicitly determined are the frustration-free spin chains with finitely correlated ground states, also known as matrix product states [13,15,40,87,99–101]. Allowing for general perturbations of such models typically leads to splitting ofthe degenerate ground states found in the frustration-free model. The so-called Kennedy triplet of‘excited’ states of the spin-1 Heisenberg antiferromagnetic chain of even length can be regarded asan example of this phenomenon [68]. In general, sufficiently small perturbations of one-dimensional


frustration-free models with a gap above the ground state will have a group of eigenvalues near thebottom of the spectrum separated by a gap (uniform in the size of the system) from the rest of thespectrum. The associated eigenstates all converge to ground states in the thermodynamic limit.Both statements are proved in [85].

We postpone a more comprehensive discussion of examples of models with distinct gapped groundstate phases until after the presentation of the stability results of gapped phases in [96].

8. Appendix

This section collects a number of facts about the decay bounds used throughout this paper. Ingeneral, we will assume that Γ is a countable set equipped with a metric, and we denote this metricby d. A good example to keep in mind is Γ = Z

ν with the ℓ1-metric. When necessary, we will alsoassume Γ is ν-regular, see (8.13).

When considering the Heisenberg dynamics associated to a Hamiltonian, our quasi-locality esti-mates require a short-range assumption on the corresponding interaction. For general sets Γ, whichneed not have the structure of a lattice, a sufficient condition for the existence of a dynamics inthe thermodynamic limit can be expressed in terms of a norm on the interaction. We have foundit convenient to express the decay of interactions with distance by a so-called F -function, whichwe discuss below. Depending on the application one has in mind, more explicit forms of decay,again expressed in terms of a family of F -functions, is convenient. These are by no means the onlyways to express decay assumptions for interaction. If generality is not the concern, one can easilyre-express decay into a more suitable form for the case at hand, say, e.g., for systems with pairinteractions only. In this appendix our goal is to briefly summarize various notions of decay whichoccur frequently in the main text.

8.1. On F -functions. Let (Γ, d) be a countable metric space. When Γ is finite, most notionsintroduced below are trivial, and for that reason we will mainly consider the situation where Γ hasinfinite cardinality. We will say that Γ is equipped with an F -function if there is a non-increasingfunction F : [0,∞) → (0,∞) for which:(i) F is uniformly integrable:

(8.1) ‖F‖ := supy∈Γ

∑

x∈Γ

F (d(x, y)) <∞

(ii) F satisfies a convolution condition:

(8.2) CF := supx,y∈Γ

∑

z∈Γ

F (d(x, z))F (d(z, y))

F (d(x, y))<∞.

Any function F satisfying (8.1) and (8.2) will be called an F -function on Γ. We note that animmediate consequence of (8.2) is that for any pair x, y ∈ Γ, we have the bound

(8.3)∑

z∈Γ

F (d(x, z))F (d(z, y)) ≤ CFF (d(x, y)).

The constant CF enters into a number of our estimates. We say that an F -function on Γ isnormalized if CF = 1. Of course, for any F -function F , the function F = C−1

F F defines a newF -function on Γ for which CF = 1.

Note that if Γ is equipped with an F -function F , then

(8.4) supx∈Γ

|bx(n)| ≤ ‖F‖F (n)−1 for any n > 0 ,


where the left-hand-side above is a uniform estimate on the cardinality of the ball of radius ncentered at x ∈ Γ. The above follows immediately from the estimate

(8.5) F (n)|bx(n)| ≤∑

y∈bx(n)

F (d(x, y)) ≤ ‖F‖.

This estimate also demonstrates that the existence of an F -function guarantees that Γ is uniformly,locally finite.

Moreover, if Γ is infinite, the existence of an F -function implies the diameter of Γ is infinite. Inthis situation, if Λnn≥1 is an increasing, exhaustive sequence of finite subsets of Γ (i.e. Λn ⊂ Λn+1

for all n ≥ 1 and Λn ↑ Γ), then for any finite X ⊂ Γ,

(8.6) d(X,Γ \ Λn) → ∞ as n→ ∞ .

This follows by observing that for any m ≥ 1,

(8.7) X(m) :=⋃

x∈X

bx(m)

is a finite subset of Γ. Since Λnn≥1 is absorbing, there is an N ≥ 1 for which X(m) ⊂ ΛN . SinceΓ has infinite cardinality, the set Γ \ΛN is non-empty. It immediately follows that d(x, y) ≥ m forall x ∈ X and y ∈ Γ \ ΛN , from which (8.6) follows.

8.1.1. Two common examples of F -functions. First, many well-studied quantum spin models aredefined on the hypercubic lattice Γ = Z

ν for some integer ν ≥ 1. For concreteness, consider Zν

equipped with the ℓ1-metric

(8.8) d(x, y) = |x− y| =ν∑

j=1

|xj − yj|.

Other translation invariant metrics can be treated similarly. For any ǫ > 0, the function

(8.9) F (r) =1

(1 + r)ν+ǫfor all r ≥ 0 ,

is an F -function on Γ = Zν . Integrability follows from

(8.10) ‖F‖ =∑

z∈Zν

1

(1 + |z|)ν+ǫ<∞.

Moreover, for any metric space (Γ, d): if p ≥ 1, the bound

(1 + d(x, y))p ≤ (1 + d(x, z) + 1 + d(z, y))p

≤ 2p−1(1 + d(x, z))p + 2p−1(1 + d(z, y))p(8.11)

holds for all x, y, z ∈ Γ, since the function t 7→ tp is (midpoint) convex. In this case, the functiondefined in (8.9) satisfies (8.2) with

(8.12) CF ≤ 2ν+ǫ‖F‖.Next, we note that for many of our results it is not necessary that Γ has the structure of a lattice.

We will say that a metric space (Γ, d) is ν-regular if there exists ν > 0 and κ <∞ for which

(8.13) supx∈Γ

|bx(n)| ≤ κnν for all n ≥ 1 .

Here for any x ∈ Γ and n ≥ 0, bx(n) is the ball of radius n centered at x and | · | denotes cardinality.From (8.4) we see that if Γ has an F -function for which F (r) ≤ Cr−ν, then Γ is ν-regular.

If (Γ, d) is ν-regular, then for any ǫ > 0, the function

(8.14) F (r) =1

(1 + r)ν+1+ǫfor all r ≥ 0,


is an F -function on Γ.To see that this is the case, we need only check uniform integrability, i.e. (8.1), as an argument

using (8.11) shows that this F satisfies (8.2). Fix x ∈ Γ. Set Bx(1) = bx(1) and Bx(n) =bx(n) \ bx(n− 1) for any n ≥ 2. It is clear then that (8.13) implies

∑

y∈Γ

1

(1 + d(x, y))ν+1+ǫ=

∞∑

n=1

∑

y∈Bx(n)

1

(1 + d(x, y))ν+1+ǫ

≤∞∑

n=1

|bx(n)|nν+1+ǫ

≤∞∑

n=1

κ

n1+ǫ<∞(8.15)

uniformly in x; hence (8.1) holds. A computation similar to (8.11) shows that CF <∞.We can combine the above discussion with (8.5) to prove the following result.

Proposition 8.1. Let (Γ, d) be a countable metric space.

(i) If (Γ, d) is ν-regular then F (r) = (1 + r)−(ν+1+ǫ) is an F function of (Γ, d) for all ǫ > 0.(ii) If F (r) = (1 + r)−(ν+ǫ) is an F -function of (Γ, d) for all ǫ > 0, then Γ is ν-regular.

Proof. Part (i) follows immediately from the discussion after (8.14). For part (ii), suppose that

F (r) = (1+ r)−(ν+ǫ) is an F -function of (Γ, d) for all ǫ > 0. Fix ǫ > 0. Then by (8.4), for any n ≥ 1and x ∈ Γ,

(8.16) |bx(n)| ≤ ‖F‖(1 + n)ν+ǫ ≤ ‖F‖(2n)ν+ǫ.

Taking the infimum over ǫ > 0 shows that (Γ, d) is ν-regular with κ = 2ν‖F‖.

8.2. On weighted F -functions. For certain applications, it is convenient to consider families ofF -functions of a specific form, which we call weighted F -functions.

Let (Γ, d) be a metric space equipped with an F -function F as described in Section 8.1. Letg : [0,∞) → [0,∞) be a non-negative, non-decreasing, sub-additive function, i.e.

(8.17) g(r + s) ≤ g(r) + g(s) for any r, s ≥ 0 .

Corresponding to any such g, the function

(8.18) Fg(r) := e−g(r)F (r) for all r ≥ 0 ,

is an F -function on Γ. In fact, since g is non-negative, Fg satisfies (8.1) with ‖Fg‖ ≤ ‖F‖. Moreover,since g is non-decreasing and sub-additive, one also has that

(8.19) g(d(x, y)) ≤ g(d(x, z) + d(z, y)) ≤ g(d(x, z)) + g(d(z, y)) for all x, y, z ∈ Γ .

Thus (8.2) holds with CFg ≤ CF .We may refer to F as the base F -function associated to Fg; note that F0 = F for g = 0 . The

function g induces a factor r 7→ e−g(r) which is often referred to as a weight. We may also looselyrefer to g as a weight and similarly Fg as a weighted F -function. One readily checks that sums,non-negative scalar multiples, and compositions of weights are also weights; in the sense that if g1and g2 are both non-negative, non-decreasing, sub-additive functions, then so too are g1 + g2, ag1(for a ≥ 0), and g1 g2.

In certain applications, it is useful to introduce a one-parameter family of weighted F -functionsby taking a base F -function F on Γ, fixing a weight g, and associating to any a ≥ 0, the functionga(r) = ag(r), for which Fga(r) = e−ag(r)F (r). When g is understood, we often just write Fa := Fga

to describe this family of weighted F -functions. For example, if F (r) = (1 + r)−p and g(r) = r,then Fa(r) = e−ar(1 + r)−p is the family of weighted F -functions defined in Section 3.1.1.


As a motivation for introducing these weights, consider again Γ = Zν . As discussed in the

previous subsection, the polynomially decaying function F in (8.9) is an F -function associated toΓ = Z

ν . Such an F -function is appropriate for interactions Φ with terms that decay polynomiallywith the diameter of their support: ‖Φ(X)‖ ≤ C(1 + diam(X))−ν+ǫ. However, we are typicallyinterested in interactions whose terms decay much faster, in particular exponentially fast. Onereadily checks that for any a > 0, the exponential function g(r) = e−ar fails to satisfy (8.2) andas such is not an F -function on Γ = Z

ν , but does satisfy the criterion to be a weight. Sinceexponential functions often govern the decay of our interactions, it is convenient that one canobtain an exponentially decaying F -function on Γ = Z

ν by making an appropriate choice of weight.Before moving on to discussing several useful weights, we point out one added benefit of these

functions. In the situation were we do not assume to have a weighted F -function and given X, Y ∈P0(Γ), we will often use the simple bound

(8.20)∑

x∈X

∑

y∈Y

F (d(x, y)) ≤ |X| ‖F‖

when applying a Lieb-Robinson bound or quasi-locality bound, see (3.24) and (5.2). For weightedF -functions, however, the following is also frequently used:

(8.21)∑

x∈X

∑

y∈Y

Fg(d(x, y)) ≤ |X| ‖F‖e−g(d(X,Y )).

Here, one typically is considering a quantum lattice system defined on a (large) finite volume Λ,and in the situation that Y = Λ \X(n) then the RHS of (8.21) decays as e−g(n).

8.2.1. Three common weights. With an eye towards our specific applications, we now introducethree particular classes of weights.

First, let µ ∈ [0, 1]. The function g : [0,∞) → [0,∞) given by g(r) = rµ is non-negative, non-decreasing, and sub-additive in the sense of (8.17). The constant function g(r) = 1 correspondingto µ = 0 is of minor interest, however, the choice of µ = 1 generates exponentially decaying weights.When 0 < µ < 1, the function e−rµ is often called a stretched exponential.

Next, we provide an example between exponential and stretched exponential decay. As we willshow, for any p > 0, the function

(8.22) g(r) =

ep

pp if 0 ≤ r ≤ ep

rln(r)p r ≥ ep

is non-negative, non-decreasing, and sub-additive. In our applications of the spectral flow, see e.g.Section 6, the choice of p = 2 is particularly relevant.

Note that at r = ep the non-constant part of g has a zero derivative, and for r > ep, g is strictlyincreasing. That motivates this particular choice of cut-off. Also, it is easy to see that this functionis sub-additive by taking cases: Let r, s ≥ 0. Consider (i) r+ s ≤ ep and (ii) r+ s > ep. Both casesare easy to see. For the second use that

(8.23) g(r + s) =r

ln(r + s)p+

s

ln(r + s)p.

Note that

(8.24) if r ≥ ep, thenr

ln(r + s)p≤ r

ln(r)pwhereas if r ≤ ep, then

r

ln(r + s)p≤ ep

ln(ep)p

the latter fact using that r + s > ep.Lastly, the function g : [0,∞) → [0,∞) given by

(8.25) g(r) = ln(1 + r)


is clearly non-negative and non-decreasing. Since for any r, s ≥ 0, we have that

(8.26) 1 + r + s ≤ (1 + r)(1 + s)

and sub-additivity of g readily follows. Starting with a base F -function, as in (8.9) or (8.14), aproper scaling of F by this weight allows for arbitrary power-law decay.

8.3. Simple transformations of F -functions. In certain applications, it is convenient to knowthat various decaying functions are in fact F -functions. Quantities of interest can be estimatedin terms of translations or re-scalings of known F -functions. For ν-regular Γ, these modificationspreserve the basic properties of an F -function. The following two propositions show that suitablydefined truncations, shifts, and dilations of F -functions are again F -functions.

Proposition 8.2. Let (Γ, d) be a ν-regular metric space with an F -function F . For any a ≥ 0 and

any choice of c ≥ F (a), the function F : [0,∞) → (0,∞) defined by setting

(8.27) F (r) =

c if 0 ≤ r ≤ a,F (r) if r > a,

is an F -function on (Γ, d). In fact,

(8.28) ‖F‖ ≤ cκaν + ‖F‖ and CF ≤ maxc, F (0) cCF

F (a)2

Proof. Fix x ∈ Γ. Note that

(8.29)∑

y∈Γ

F (d(x, y)) = c|bx(a)| +∑

y∈Γ\bx(a)

F (d(x, y))

and the first bound in (8.28) follows from ν-regularity.To see the second bound, note that

(8.30) F (r) ≤ c

F (a)F (r) ⇒

∑

z∈Γ

F (d(x, z))F (d(z, y)) ≤(

c

F (a)

)2

CFF (d(x, y))

for all sites x, y ∈ Γ. By considering the cases d(x, y) ≥ a and d(x, y) < a separately, one can show

cF (d(x, y)) ≤ maxc, F (0)F (d(x, y)),

from which the second bound in (8.28) follows.

Proposition 8.3. Let (Γ, d) be a ν-regular metric space, and p ≥ 1 be such that

(8.31) F0(r) =1

(1 + r)p, for all r ≥ 0

is an F -function on (Γ, d).

(i) If F (r) = e−g(r)F0(r) is a weighted F -function on (Γ, d), then for any ǫ > 0 the function

defined by F (r) = F (ǫr) is an F -function on (Γ, d). Moreover,

(8.32) ‖F‖ ≤ max1, ǫ−p‖F0‖ and CF ≤ 2p‖F‖ .(ii) If F (r) = e−g(r)F0(r) is a weighted F -function on (Γ, d) then for any a > 0 the function

defined by

(8.33) F (r) =

F (0) 0 ≤ r ≤ a,F (r − a) r > a,

is an F -function on (Γ, d). In fact,

(8.34) ‖F‖ ≤ max1, F (0)F (a)

‖F‖ and CF ≤(

max1, F (0)F (a)

)2

CF .


Proof. To see (i), first note that r 7→ g(ǫr) is non-negative, non-decreasing, and sub-additive. Inthis case, we need only verify that r 7→ (1 + ǫr)−p is an F -function. For any ǫ > 0, one has that

(8.35) min1, ǫ(1 + r) ≤ (1 + ǫr) ≤ max1, ǫ(1 + r)

holds for all r ≥ 0. The first bound in (8.32) is then clear, and the second bound follows as in(8.11).

To see (ii), a short calculation shows that

(8.36) F (r) ≤ F (r) ≤ max1, F (0)F (a)

F (r) for all r ≥ 0 .

The first inequality above is trivial since F is non-increasing. The second follows from sub-additivityof g, namely g(r) ≤ g(a) + g(r− a) for any r > a, as well as the fact that F0(r− a) ≤ F0(r)/F0(a).The bounds in (8.34) readily follow.

A simple consequence of Proposition 8.3 is the following. Under the assumptions of Proposi-tion 8.3, let F be a weighted F -function with base F0. For any r > 1, the bound

(8.37) F (r) ≤ F (⌊r⌋) ≤ F (r − 1)

is clear, and this bound extends to all r ≥ 0 using the function in Proposition 8.3 (ii) above. ThusF (⌊r⌋) is an F -function as well.

8.4. Basic interaction bounds. In the main text, we frequently use a number of basic estimatesconcerning interactions that are expressed using F -functions. Here we collect a few results for laterreference.

We begin by recalling some of the basic notation associated with interactions. Let (Γ, d) be ametric space equipped with an F -function F as discussed in Section 8.1. Let P0(Γ) denote the setof finite subsets of Γ. We say that a mapping Φ is an interaction on Γ if Φ : P0(Γ) → Aloc

Γ withthe property that Φ(X)∗ = Φ(X) ∈ AX for every X ∈ P0(Γ). If Φ is an interaction on Γ, we writethat Φ ∈ BF if and only if

(8.38) ‖Φ‖F := supx,y∈Γ

1

F (d(x, y))

∑

X∈P0(Γ):

x,y∈X

‖Φ(X)‖ <∞.

A basic consequence of (8.38) is that for all x, y ∈ Γ,

(8.39)∑

X∈P0(Γ):x,y∈X

‖Φ(X)‖ ≤ ‖Φ‖FF (d(x, y)).

8.4.1. Estimates based on distance. We first provide a basic F -norm estimate based on summinginteraction terms whose distance from a specific set is given. Recall that if X ⊂ Γ and n ≥ 0, theset X(n) ⊂ Γ is defined as

(8.40) X(n) = y ∈ Γ : d(X, y) ≤ n =⋃

x∈X

bx(n).

Proposition 8.4. Let (Γ, d) be a metric space equipped with an F -function F . Let Φ be an inter-action on Γ with Φ ∈ BF . For any X ∈ P0(Γ) and each R ≥ 0,

(8.41)∑

Z∈P0(Γ):

d(Z,X)≤R

‖Φ(Z)‖ ≤ ‖Φ‖F∑

x∈X(R)

∑

y∈Γ

F (d(x, y)) ≤ |X(R)|‖F‖‖Φ‖F .


Moreover,

(8.42)∑

Z∈P0(Γ):

d(Z,X)>R

‖Φ(Z)‖∑

x∈X

∑

z∈Z

F (d(x, z)) ≤ CF ‖Φ‖F∑

x∈X

∑

y∈Γ\X(R)

F (d(x, y)).

Proof. Given the uniform integrability of F , i.e. (8.1), the second estimate in (8.41) is clear giventhe first. To see the first, note that by over-counting,

(8.43)∑

Z∈P0(Γ):

d(Z,X)≤R

‖Φ(Z)‖ ≤∑

x∈X(R)

∑

y∈Γ

∑

Z∈P0(Γ):

x,y∈Z

‖Φ(Z)‖.

The estimate in (8.41) now follows from (8.39) and (8.1).To prove (8.42), note that again by over-counting,

∑

Z∈P0(Γ):

d(Z,X)>R

‖Φ(Z)‖∑

x∈X

∑

z∈Z

F (d(x, z)) ≤∑

x∈X

∑

z∈Γ

F (d(x, z))∑

y∈Γ\X(R)

∑

Z∈P0(Γ):

z,y∈Z

‖Φ(Z)‖

≤ ‖Φ‖F∑

x∈X

∑

y∈Γ\X(R)

∑

z∈Γ

F (d(x, z))F (d(z, y)).(8.44)

Thus, (8.42) follows using the convolution condition on F .

A simple corollary of these bounds follows. To state it requires that we introduce two notions.First, we describe compatible F -functions. Let (Γ, d) be a metric space equipped with two F -functions denoted by F1 and F2. We will say that F1 and F2 are compatible if there is a positivenumber C1,2 and a non-increasing function F1,2 : [0,∞) → (0,∞) for which: given any x, y ∈ Γ,

(8.45)∑

z∈Γ

F1(d(x, z))F2(d(z, y)) ≤ C1,2F1,2(d(x, y)).

Next we briefly describe time-dependent interactions (see Section 3.1.1 for more details). Let I ⊂ R

be an interval. We say that Φ : P0(Γ) × I → AlocΓ is a strongly continuous interaction on Γ if for

each X ∈ P0(Γ):

(i) Φ(X, t)∗ = Φ(X, t) ∈ AX for all t ∈ I;(ii) The map Φ(X, ·) : I → AX is continuous in the strong operator topology.

Moreover, we will write Φ ∈ BF (I) if Φ(·, t) ∈ BF , for all t ∈ I, and ‖Φ(t)‖F , which we sometimeswrite as ‖Φ‖F (t), is a locally bounded function of t.

We now state a corollary of Proposition 8.4.

Corollary 8.5. Let (Γ, d) be a metric space equipped with two compatible F -functions F1 and F2.For an interval I ⊂ R, suppose there is a co-cycle of automorphisms αt,st,s∈I of AΓ, which satisfya Lieb-Robinson bound, i.e. for any disjoint X,Y ∈ P0(Γ), each A ∈ AX , B ∈ AY , and s, t ∈ I,

(8.46) ‖[αt,s(A), B]‖ ≤ 2‖A‖‖B‖CF1

Dα(t, s)∑

x∈X

∑

y∈Y

F1(d(x, y))

with a pre-factor Dα(t, s) that increases as |t− s| increases. Let Ψ be a time-dependent interactionwith Ψ ∈ BF2(I). Given R ≥ 0 and s, t ∈ I with s ≤ t, one has that for any A ∈ AX withX ∈ P0(Γ),

(8.47)∑

Z∈P0(Γ):

d(Z,X)≤R

∫ t

s‖[αt,r(A),Ψ(Z, r)]‖ dr ≤ 2‖A‖

∫ t

s‖Ψ‖F2(r) dr

∑

x∈X(R)

∑

y∈Γ

F2(d(x, y))


and,

∑

Z∈P0(Γ):

d(Z,X)>R

∫ t

s‖[αt,r(A),Ψ(Z, r)]‖ dr ≤

2‖A‖C1,2

CF1

Dα(t, s)

∫ t

s‖Ψ‖F2(r) dr

∑

x∈X

∑

y∈Γ\X(R)

F1,2(d(x, y)).(8.48)

Proof. To prove (8.47), note that for any s ≤ r ≤ t, the simple bound

(8.49) ‖[αt,r(A),Ψ(Z, r)]‖ ≤ 2‖A‖‖Ψ(Z, r)‖holds. Estimating as in (8.41) yields (8.47) as claimed.

To see (8.48), take some s ≤ r ≤ t and note that (8.46) applies. In fact, applying the Lieb-Robinson bound we find

(8.50) ‖[αt,r(A),Ψ(Z, r)]‖ ≤ 2‖A‖‖Ψ(Z, r)‖CF1

Dα(t, s)∑

x∈X

∑

z∈Z

F1(d(x, z)).

Here, we use that, by our assumptions, Dα(t, r) ≤ Dα(t, s). Then, arguing as in the proof of (8.41),see in particular (8.44), we find

(8.51)∑

Z∈P0(Γ):

d(Z,X)>R

‖Ψ(Z, r)‖∑

x∈X

∑

z∈Z

F1(d(x, z)) ≤ ‖Ψ‖F2(r)∑

x∈X

∑

y∈Γ\X(R)

∑

z∈Γ

F1(d(x, z))F2(d(z, y)).

Using compatibility, i.e. (8.45), the bound in (8.48) now follows upon integration.

8.4.2. Estimates based on diameter. In some situations, it is more convenient to form decay argu-ments based on the diameters of sets, rather than the distance between sets. For these cases, wewill further assume that (Γ, d) is ν-regular so that (8.13) holds.

Before we state our first result, let us introduce some convenient notation. Our estimates willoften be in terms of moments of certain decay functions. To this end, let G : [0,∞) → (0,∞) be adecay function. For any p ≥ 0 and each m ≥ 0, set

(8.52) MGp (m) =

∞∑

n=⌊m⌋

(1 + n)pG(n)

whenever the sum on the right-hand-side above is finite. We will refer toMGp (0) as the p-th moment

of G. The notation

(8.53) MGp MG

q (m) =∞∑

n=⌊m⌋

(1 + n)p∞∑

n′=n

(1 + n′)qG(n′)

will be used for iterated moments. A rough estimate involving exchanging the order of the sum-mations shows

(8.54) MGp MG

q (m) ≤MGp+q+1(m).

We now state our first result, compare with Proposition 8.4.

Proposition 8.6. Let (Γ, d) be a ν-regular metric space equipped with an F -function F , and Φ bean interaction on Γ with Φ ∈ BF . Then, for any R ≥ 0,

(8.55) supx∈Γ

∑

Z∈P0(Γ):

x∈Z;diam(Z)≤R

‖Φ(Z)‖ ≤ ‖Φ‖F ‖F‖.


If, in addition, F has a finite 2ν-th moment, i.e. MF2ν(0) <∞, then

(8.56) supx∈Γ

∑

Z∈P0(Γ):

x∈Z;diam(Z)>R

‖Φ(Z)‖ ≤ κ2‖Φ‖FMF2ν(R).

Proof. For any fixed x ∈ Γ, note that

(8.57)∑

Z∈P0(Γ):

x∈Z;diam(Z)≤R

‖Φ(Z)‖ ≤∑

y∈bx(R)

∑

Z∈P0(Γ):

x,y∈Z

‖Φ(Z)‖ ≤ ‖Φ‖F∑

y∈bx(R)

F (d(x, y)).

Using this bound, (8.55) follows from uniform integrability of F .To see (8.56), again fix x ∈ Γ and see that

∑

Z∈P0(Γ):x∈Z; diam(Z)>R

‖Φ(Z)‖ ≤∞∑

n=⌊R⌋

∑

Z∈P0(Γ):x∈Z;n≤diam(Z)<n+1

‖Φ(Z)‖

≤∞∑

n=⌊R⌋

∑

w,z∈bx(n+1):n≤d(w,z)<n+1

∑

Z∈P0(Γ):w,z∈Z

‖Φ(Z)‖

≤ ‖Φ‖F∞∑

n=⌊R⌋

∑


F (d(w, z))

≤ κ2‖Φ‖F∞∑

n=⌊R⌋

(1 + n)2νF (n)(8.58)

where, for the last line above, we used that F is non-increasing and over-estimated using (8.13).This proves (8.56).

In some arguments, we encounter moments of interactions. More precisely, let Φ be an interactionon Γ. For any p ≥ 0, the mapping Φp : P0(Γ) → Aloc

Γ given by

(8.59) Φp(X) = |X|pΦ(X)

also defines an interaction on Γ. We refer to Φp as the p-th moment of Φ. The next lemma providesa basic estimate for interactions of this type.

Lemma 8.7. Let (Γ, d) be a ν-regular metric space equipped with an F -function F , and Φ be aninteraction on Γ with Φ ∈ BF . If p ≥ 0 and MF

(p+2)ν(0) <∞, then the p-th moment of Φ satisfies

(8.60)∑

X∈P0(Γ):x,y∈X

‖Φp(X)‖ ≤ κp+2‖Φ‖FMF(p+2)ν(d(x, y)) .

Proof. Fix x, y ∈ Γ. Set m = ⌊d(x, y)⌋ and note that

(8.61)∑

X∈P0(Γ):x,y∈X

‖Φp(X)‖ ≤∑

n≥m

∑

X∈P0(Γ):x,y∈Xn≤diam(X)<n+1

|X|p‖Φ(X)‖

Now, if x ∈ X and diam(X) < n+ 1, then (8.13) guarantees that

(8.62) |X| ≤ |bx(n+ 1)| ≤ κ(n + 1)ν and therefore |X|p ≤ κp(n+ 1)pν


Moreover, by over-counting, it is clear that

(8.63)∑

X∈P0(Γ):x,y∈Xn≤diam(X)<n+1

∗ ≤∑

w,z∈bx(n+1)n≤d(w,z)<n+1

∑

X∈P0(Γ):w,z∈X

∗

where ∗ represents a non-negative quantity. We conclude that∑

X∈P0(Γ):x,y∈X

‖Φp(X)‖ ≤ κp∑

n≥m

(n+ 1)pν∑


∑

X∈P0(Γ):z,w∈X

‖Φ(X)‖

≤ κp‖Φ‖F∑

n≥m

(n + 1)pν∑


F (d(w, z))

≤ κp+2‖Φ‖F∑

n≥m

(n+ 1)(p+2)νF (n) ,(8.64)

which proves (8.60).

When considering a weighted F -function F (r) = e−g(r)F0(r), one can often use the weight with(8.60) to prove that the p-th moment of Φ has a finite F -norm. This allows us to apply theLieb-Robinson bound theory to Φp.

Corollary 8.8. Let p ≥ 0 and F (r) = e−g(r)F0(r) be a weighted F -function on a ν-regular metric

space (Γ, d). If M−ag(p+2)ν(0) <∞ for some 0 < a ≤ 1, then Φp ∈ BF for any Φ ∈ BF where F is the

F -function

(8.65) F (r) = e(a−1)g(⌊r⌋)F0(⌊r⌋).Proof. This is an immediate consequence of (8.60), and the bound

MF(p+2)ν(k) =

∞∑

n=⌊k⌋

(1 + n)(p+2)νe−g(n)F0(n)

≤ e(a−1)g(⌊k⌋)F0(⌊k⌋)∞∑

n=0

(1 + n)(p+2)νe−ag(n)

= M−ag(p+2)ν(0)F (k)

for all k ≥ 0.

8.4.3. An estimate on weighted sums.

Lemma 8.9. Let (Γ, d) be a ν-regular metric space equipped with an F -function F , and Φ be aninteraction on Γ with Φ ∈ BF . If G : [0,∞) → (0,∞) satisfies MG

2ν+1(0) <∞, then for any x, y ∈ Γ

(8.66)

∞∑

n=0

G(n)∑

X∈P0(Γ):x,y∈X(n+1)

‖Φ(X)‖ ≤ κ‖Φ‖F(

κMG2ν+1(0)F (d(x, y)/3) + ‖F‖MG

ν+1(d(x, y)/3))

.

Proof. For each X ∈ P0(Γ), the sets X(n), see e.g. (8.40), are increasing and therefore, if x, y ∈X(n) for some n ≥ 1, then x, y ∈ X(m) for all m ≥ n. With this in mind, we write

(8.67)∞∑

n=0

G(n)∑

X∈P0(Γ):x,y∈X(n+1)

‖Φ(X)‖ =∞∑

m=0

∑

X∈P0(Γ):x,y∈X(m+1)

χm+1(X)‖Φ(X)‖∞∑

n=m

G(n)


where we have inserted the characteristic function χm+1 defined by

(8.68) χm+1(X) =

0 if x, y ∈ X(m),1 otherwise.

Note that x, y ∈ X(m+1) means there exist w, z ∈ X such that w ∈ bx(m+1) and z ∈ by(m+1).Using this, the definition of MG

0 (m) (see (8.52)), and the F -norm of Φ, the estimate

∞∑

n=0

G(n)∑

X∈P0(Γ):x,y∈X(n+1)

‖Φ(X)‖ ≤∞∑

m=0

MG0 (m)

∑

w∈bx(m+1)z∈by(m+1)

∑

X∈P0(Γ):w,z∈X

‖Φ(X)‖

≤ ‖Φ‖F∞∑

m=0

MG0 (m)

∑


F (d(w, z))(8.69)

readily follows.We now divide the final series on the right-hand-side of (8.69) into sums of large and small m.

More precisely, for any fixed 0 < ǫ < 1, set m0 = m0(ǫ, x, y) to be the largest integer m ≥ −1satisfying

(8.70) 2(m+ 1) ≤ (1− ǫ)d(x, y) .

For any 0 ≤ m ≤ m0, w ∈ bx(m+ 1) and z ∈ by(m+ 1), on has ǫd(x, y) ≤ d(w, z) as

d(x, y) ≤ d(x,w) + d(w, z) + d(z, y) ≤ 2(m+ 1) + d(w, z)

≤ (1− ǫ)d(x, y) + d(w, z).(8.71)

In this case, the first few terms may be estimated asm0∑

m=0

MG0 (m)

∑


F (d(w, z)) ≤ κ2F (ǫd(x, y))

m0∑

m=0

(1 +m)2νMG0 (m)

≤ κ2MG2ν+1(0)F (ǫd(x, y))(8.72)

where in the last equality we have used (8.54).For the remaining terms, uniform integrability of F implies

(8.73)∑

m≥m0+1

MG0 (m)

∑


F (d(w, z)) ≤ κ‖F‖MGν MG

0 (m0 + 1).

Using the definition of m0 and again applying (8.54), the bound claimed in (8.66) follows fromchoosing ǫ = 1/3.

Acknowledgements

We would like to thank Valentin Zagrebnov for illuminating discussions about the non-autonomousCauchy problems that arise in quantum dynamical systems. We also thank Martin Gebert for read-ing an early version of this paper and asking good questions and Derek Robinson for several usefulremarks and informative comments about the history of the subject. All three authors wish tothank the Departments of Mathematics of the University of Arizona and the University of Cali-fornia, Davis, for extending their kind hospitality to us and for the stimulating atmosphere theyoffered during several visits back and forth over the years it took to complete this project. BN alsoacknowledges the support of a CRM-Simons Professorship for a stay at the Centre de RecherchesMathematiques (Montreal) during Fall 2018, which created the perfect circumstances to completethis paper.


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[130] A. Young, Spectral properties of multi-dimensional quantum spin systems, Ph.D. thesis, University of California,Davis, 2016.

Department of Mathematics and Center for Quantum Mathematics and Physics, University of

California, Davis, Davis, CA 95616, USA

E-mail address: [email protected]

Department of Mathematics, University of Arizona, Tuscon, AZ 85721, USA


Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA


abstract. arxiv:1810.02428v2 [math-ph] 1 mar 2019 · 3. lieb-robinson bounds and inﬁnite volume...

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